Introduction and Definitions
The Result
An exact algorithm for Subset Feedback Vertex
Set on chordal graphs
Petr A. Golovach1
1
Pinar Heggernes1
Reza Saei1
Dieter Kratsch2
Department of Informatics, University of Bergen, Norway.
2 LITA, Université de Lorraine - Metz, France.
IPEC 2012
September 12-14, 2012
Ljubljana, Slovenia
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition
Given a graph G = (V , E ) and a set S ⊆ V , a set U ⊆ V is a
subset feedback vertex set (sfvs) of (G , S) if no cycle in G [V \ U]
contains a vertex of S.
•
•
V
Is
Vertices in S
Vertices in U
\ U is said to be an S-forest.
any proper subset of U a sfvs for (G , S)?
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition
Given a graph G = (V , E ) and a set S ⊆ V , a set U ⊆ V is a
subset feedback vertex set (sfvs) of (G , S) if no cycle in G [V \ U]
contains a vertex of S.
•
•
V
Is
Vertices in S
Vertices in U
\ U is said to be an S-forest.
any proper subset of U a sfvs for (G , S)?
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition
Given a graph G = (V , E ) and a set S ⊆ V , a set U ⊆ V is a
subset feedback vertex set (sfvs) of (G , S) if no cycle in G [V \ U]
contains a vertex of S.
•
•
V
Is
Vertices in S
Vertices in U
\ U is said to be an S-forest.
any proper subset of U a sfvs for (G , S)?
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition
Given a graph G = (V , E ) and a set S ⊆ V , a set U ⊆ V is a
subset feedback vertex set (sfvs) of (G , S) if no cycle in G [V \ U]
contains a vertex of S.
•
•
V
Is
Vertices in S
Vertices in U
\ U is said to be an S-forest.
any proper subset of U a sfvs for (G , S)?
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition
Given a graph G = (V , E ) and a set S ⊆ V , a set U ⊆ V is a
subset feedback vertex set (sfvs) of (G , S) if no cycle in G [V \ U]
contains a vertex of S.
•
•
V
Is
Vertices in S
Vertices in U
\ U is said to be an S-forest.
any proper subset of U a sfvs for (G , S)?
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition
Given a graph G = (V , E ) and a set S ⊆ V , a set U ⊆ V is a
subset feedback vertex set (sfvs) of (G , S) if no cycle in G [V \ U]
contains a vertex of S.
•
•
V
Is
Vertices in S
Vertices in U
\ U is said to be an S-forest.
any proper subset of U a sfvs for (G , S)?
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition
Given a graph G = (V , E ) and a set S ⊆ V , a set U ⊆ V is a
subset feedback vertex set (sfvs) of (G , S) if no cycle in G [V \ U]
contains a vertex of S.
•
•
V
Is
Vertices in S
Vertices in U
\ U is said to be an S-forest.
any proper subset of U a sfvs for (G , S)?
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition
Given a graph G = (V , E ) and a set S ⊆ V , a set U ⊆ V is a
subset feedback vertex set (sfvs) of (G , S) if no cycle in G [V \ U]
contains a vertex of S.
•
•
V
Is
Vertices in S
Vertices in U
\ U is said to be an S-forest.
any proper subset of U a sfvs for (G , S)?
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition
Given a graph G = (V , E ) and a set S ⊆ V , a set U ⊆ V is a
subset feedback vertex set (sfvs) of (G , S) if no cycle in G [V \ U]
contains a vertex of S.
•
•
V
Is
Vertices in S
Vertices in U
\ U is said to be an S-forest.
any proper subset of U a sfvs for (G , S)?
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition
Given a graph G = (V , E ) and a set S ⊆ V , a set U ⊆ V is a
subset feedback vertex set (sfvs) of (G , S) if no cycle in G [V \ U]
contains a vertex of S.
•
•
V
Is
Vertices in S
Vertices in U
\ U is said to be an S-forest.
any proper subset of U a sfvs for (G , S)?
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition
Given a graph G = (V , E ) and a set S ⊆ V , a set U ⊆ V is a
subset feedback vertex set (sfvs) of (G , S) if no cycle in G [V \ U]
contains a vertex of S.
•
•
V
Is
Vertices in S
Vertices in U
\ U is said to be an S-forest.
any proper subset of U a sfvs for (G , S)?
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition
Given a graph G = (V , E ) and a set S ⊆ V , a set U ⊆ V is a
subset feedback vertex set (sfvs) of (G , S) if no cycle in G [V \ U]
contains a vertex of S.
•
•
V
Is
Vertices in S
Vertices in U
\ U is said to be an S-forest.
any proper subset of U a sfvs for (G , S)?
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
U is minimal sfvs if no sfvs of (G , S) is a proper subset of U.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
U is minimal sfvs if no sfvs of (G , S) is a proper subset of U.
When U is a minimal sfvs of (G , S) then V \ U is a maximal
S-forest of G .
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
U is minimal sfvs if no sfvs of (G , S) is a proper subset of U.
When U is a minimal sfvs of (G , S) then V \ U is a maximal
S-forest of G .
Every sfvs of minimum cardinality or minimum weight is a
minimal sfvs.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Subset Feedback Vertex Set (SFVS)
Input: G , S and an integer k
Question: Does (G , S) have a sfvs of size ≤ k?
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Subset Feedback Vertex Set (SFVS)
Input: G , S and an integer k
Question: Does (G , S) have a sfvs of size ≤ k?
Weighted Subset Feedback Vertex Set (WSFVS)
Input: Weighted graph G , S and a weight ω
Question: Does (G , S) have a sfvs of total weight ≤ ω?
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Subset Feedback Vertex Set (SFVS)
Input: G , S and an integer k
Question: Does (G , S) have a sfvs of size ≤ k?
Weighted Subset Feedback Vertex Set (WSFVS)
Input: Weighted graph G , S and a weight ω
Question: Does (G , S) have a sfvs of total weight ≤ ω?
SFVS was introduced by [Even, Naor and Zosin - 2000] and it
generalizes several well-studied problems.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Subset Feedback Vertex Set (SFVS)
Input: G , S and an integer k
Question: Does (G , S) have a sfvs of size ≤ k?
Weighted Subset Feedback Vertex Set (WSFVS)
Input: Weighted graph G , S and a weight ω
Question: Does (G , S) have a sfvs of total weight ≤ ω?
SFVS was introduced by [Even, Naor and Zosin - 2000] and it
generalizes several well-studied problems.
1
When S = V , it is equivalent to the classical Feedback
Vertex Set (FVS) problem.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Subset Feedback Vertex Set (SFVS)
Input: G , S and an integer k
Question: Does (G , S) have a sfvs of size ≤ k?
Weighted Subset Feedback Vertex Set (WSFVS)
Input: Weighted graph G , S and a weight ω
Question: Does (G , S) have a sfvs of total weight ≤ ω?
SFVS was introduced by [Even, Naor and Zosin - 2000] and it
generalizes several well-studied problems.
1
When S = V , it is equivalent to the classical Feedback
Vertex Set (FVS) problem.
2
When |S| = 1, it generalizes the Multiway Cut problem.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
What is done on the problem for general graphs so far?
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
What is done on the problem for general graphs so far?
1
Weighted subset feedback vertex set admits a
polynomial-time 8-approximation algorithm.
[Even, Naor and Zosin - 2000]
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
What is done on the problem for general graphs so far?
1
Weighted subset feedback vertex set admits a
polynomial-time 8-approximation algorithm.
[Even, Naor and Zosin - 2000]
2
The unweighted version of the problem is fixed parameter
tractable when parameterized by k.
[Cygan, Pilipczuk, Pilipczuk and Wojtaszczyk - 2011]
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
What is done on the problem for general graphs so far?
1
Weighted subset feedback vertex set admits a
polynomial-time 8-approximation algorithm.
[Even, Naor and Zosin - 2000]
2
The unweighted version of the problem is fixed parameter
tractable when parameterized by k.
[Cygan, Pilipczuk, Pilipczuk and Wojtaszczyk - 2011]
3
The only exact algorithm known for its weighted version runs
in O(1.8638n ) time and solves the problem by enumerating all
minimal sfvs.
[Fomin, Heggernes, Kratsch, Papadopoulos and Villanger - 2011]
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
FVS has been studied on many graph classes like AT-free
graphs and chordal graphs and several positive results exist.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
FVS has been studied on many graph classes like AT-free
graphs and chordal graphs and several positive results exist.
FVS is polynomial time solvable on chordal graphs, but
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
FVS has been studied on many graph classes like AT-free
graphs and chordal graphs and several positive results exist.
FVS is polynomial time solvable on chordal graphs, but
SFVS is NP-complete on chordal graphs and even on their
restricted subclass split graphs.
[Fomin, Heggernes, Kratsch, Papadopoulos and Villanger - 2011]
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
FVS has been studied on many graph classes like AT-free
graphs and chordal graphs and several positive results exist.
FVS is polynomial time solvable on chordal graphs, but
SFVS is NP-complete on chordal graphs and even on their
restricted subclass split graphs.
[Fomin, Heggernes, Kratsch, Papadopoulos and Villanger - 2011]
We give an algorithm for WSFVS with running time
O(1.6708n ) when the input graph is chordal.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition (Chordal graph)
A graph is chordal if every cycle of length at least 4 has a chord.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition (Chordal graph)
A graph is chordal if every cycle of length at least 4 has a chord.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition (Chordal graph)
A graph is chordal if every cycle of length at least 4 has a chord.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition (Chordal graph)
A graph is chordal if every cycle of length at least 4 has a chord.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition (Simplicial vertex)
A vertex v of G is called simplicial in G if N(v ) is a clique.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition (Simplicial vertex)
A vertex v of G is called simplicial in G if N(v ) is a clique.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition (Simplicial vertex)
A vertex v of G is called simplicial in G if N(v ) is a clique.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition (Simplicial vertex)
A vertex v of G is called simplicial in G if N(v ) is a clique.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition (perfect elimination order)
An ordering v1 · · · vn of the vertices of G is a perfect elimination
order of G if for all i, vi is simplicial in G [vi · · · vn ].
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition (perfect elimination order)
An ordering v1 · · · vn of the vertices of G is a perfect elimination
order of G if for all i, vi is simplicial in G [vi · · · vn ].
A graph is chordal if and only if it has a perfect elimination order.
[Fulkerson and Gross 1965]
Every induced subgraph of a chordal graph is chordal.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition (perfect elimination order)
An ordering v1 · · · vn of the vertices of G is a perfect elimination
order of G if for all i, vi is simplicial in G [vi · · · vn ].
A graph is chordal if and only if it has a perfect elimination order.
[Fulkerson and Gross 1965]
Every induced subgraph of a chordal graph is chordal.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition (perfect elimination order)
An ordering v1 · · · vn of the vertices of G is a perfect elimination
order of G if for all i, vi is simplicial in G [vi · · · vn ].
A graph is chordal if and only if it has a perfect elimination order.
[Fulkerson and Gross 1965]
Every induced subgraph of a chordal graph is chordal.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition (perfect elimination order)
An ordering v1 · · · vn of the vertices of G is a perfect elimination
order of G if for all i, vi is simplicial in G [vi · · · vn ].
A graph is chordal if and only if it has a perfect elimination order.
[Fulkerson and Gross 1965]
Every induced subgraph of a chordal graph is chordal.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition (perfect elimination order)
An ordering v1 · · · vn of the vertices of G is a perfect elimination
order of G if for all i, vi is simplicial in G [vi · · · vn ].
A graph is chordal if and only if it has a perfect elimination order.
[Fulkerson and Gross 1965]
Every induced subgraph of a chordal graph is chordal.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition (perfect elimination order)
An ordering v1 · · · vn of the vertices of G is a perfect elimination
order of G if for all i, vi is simplicial in G [vi · · · vn ].
A graph is chordal if and only if it has a perfect elimination order.
[Fulkerson and Gross 1965]
Every induced subgraph of a chordal graph is chordal.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition (perfect elimination order)
An ordering v1 · · · vn of the vertices of G is a perfect elimination
order of G if for all i, vi is simplicial in G [vi · · · vn ].
A graph is chordal if and only if it has a perfect elimination order.
[Fulkerson and Gross 1965]
Every induced subgraph of a chordal graph is chordal.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition (perfect elimination order)
An ordering v1 · · · vn of the vertices of G is a perfect elimination
order of G if for all i, vi is simplicial in G [vi · · · vn ].
A graph is chordal if and only if it has a perfect elimination order.
[Fulkerson and Gross 1965]
Every induced subgraph of a chordal graph is chordal.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition (perfect elimination order)
An ordering v1 · · · vn of the vertices of G is a perfect elimination
order of G if for all i, vi is simplicial in G [vi · · · vn ].
A graph is chordal if and only if it has a perfect elimination order.
[Fulkerson and Gross 1965]
Every induced subgraph of a chordal graph is chordal.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition (perfect elimination order)
An ordering v1 · · · vn of the vertices of G is a perfect elimination
order of G if for all i, vi is simplicial in G [vi · · · vn ].
A graph is chordal if and only if it has a perfect elimination order.
[Fulkerson and Gross 1965]
Every induced subgraph of a chordal graph is chordal.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition (perfect elimination order)
An ordering v1 · · · vn of the vertices of G is a perfect elimination
order of G if for all i, vi is simplicial in G [vi · · · vn ].
A graph is chordal if and only if it has a perfect elimination order.
[Fulkerson and Gross 1965]
Every induced subgraph of a chordal graph is chordal.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition (perfect elimination order)
An ordering v1 · · · vn of the vertices of G is a perfect elimination
order of G if for all i, vi is simplicial in G [vi · · · vn ].
A graph is chordal if and only if it has a perfect elimination order.
[Fulkerson and Gross 1965]
Every induced subgraph of a chordal graph is chordal.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition (perfect elimination order)
An ordering v1 · · · vn of the vertices of G is a perfect elimination
order of G if for all i, vi is simplicial in G [vi · · · vn ].
A graph is chordal if and only if it has a perfect elimination order.
[Fulkerson and Gross 1965]
Every induced subgraph of a chordal graph is chordal.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition (perfect elimination order)
An ordering v1 · · · vn of the vertices of G is a perfect elimination
order of G if for all i, vi is simplicial in G [vi · · · vn ].
A graph is chordal if and only if it has a perfect elimination order.
[Fulkerson and Gross 1965]
Every induced subgraph of a chordal graph is chordal.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition (perfect elimination order)
An ordering v1 · · · vn of the vertices of G is a perfect elimination
order of G if for all i, vi is simplicial in G [vi · · · vn ].
A graph is chordal if and only if it has a perfect elimination order.
[Fulkerson and Gross 1965]
Every induced subgraph of a chordal graph is chordal.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition (perfect elimination order)
An ordering v1 · · · vn of the vertices of G is a perfect elimination
order of G if for all i, vi is simplicial in G [vi · · · vn ].
A graph is chordal if and only if it has a perfect elimination order.
[Fulkerson and Gross 1965]
Every induced subgraph of a chordal graph is chordal.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition (perfect elimination order)
An ordering v1 · · · vn of the vertices of G is a perfect elimination
order of G if for all i, vi is simplicial in G [vi · · · vn ].
A graph is chordal if and only if it has a perfect elimination order.
[Fulkerson and Gross 1965]
Every induced subgraph of a chordal graph is chordal.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Definition (perfect elimination order)
An ordering v1 · · · vn of the vertices of G is a perfect elimination
order of G if for all i, vi is simplicial in G [vi · · · vn ].
A graph is chordal if and only if it has a perfect elimination order.
[Fulkerson and Gross 1965]
Every induced subgraph of a chordal graph is chordal.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
We present a branching algorithm that lists all minimal sfvs of
chordal graphs.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
We present a branching algorithm that lists all minimal sfvs of
chordal graphs.
The algorithm gives also an upper bound on the maximum
number of minimal sfvs on chordal graphs.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
On the maximum number of minimal sfvs we have,
General graphs
Chordal graphs
Lower bound
1.5927n [1]
1.5848n [3]
Upper bound
1.8638n [2]
1. [Fomin, Gaspers, Kratsch, Pyatkin and Razgon - 2008]
2. [Fomin, Heggernes, Kratsch, Papadopoulos and Villanger - 2011]
3. [Couturier, Heggernes, Van ’t Hof and Villanger - 2012]
Our result tightens the gap between the upper and lower bounds
on chordal graphs.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
On the maximum number of minimal sfvs we have,
General graphs
Chordal graphs
Lower bound
1.5927n [1]
1.5848n [3]
Upper bound
1.8638n [2]
1.6708n
1. [Fomin, Gaspers, Kratsch, Pyatkin and Razgon - 2008]
2. [Fomin, Heggernes, Kratsch, Papadopoulos and Villanger - 2011]
3. [Couturier, Heggernes, Van ’t Hof and Villanger - 2012]
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
On the maximum number of minimal sfvs we have,
General graphs
Chordal graphs
Lower bound
1.5927n [1]
1.5848n [3]
Upper bound
1.8638n [2]
1.6708n
1. [Fomin, Gaspers, Kratsch, Pyatkin and Razgon - 2008]
2. [Fomin, Heggernes, Kratsch, Papadopoulos and Villanger - 2011]
3. [Couturier, Heggernes, Van ’t Hof and Villanger - 2012]
Our result tightens the gap between the upper and lower bounds
on chordal graphs.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Theorem
All minimal sfvs of a chordal graph on n vertices can be listed in
O(1.6708n ) time.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Theorem
All minimal sfvs of a chordal graph on n vertices can be listed in
O(1.6708n ) time.
Corollary
A chordal graph on n vertices has at most 1.6708n minimal sfvs.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Theorem
All minimal sfvs of a chordal graph on n vertices can be listed in
O(1.6708n ) time.
Corollary
A chordal graph on n vertices has at most 1.6708n minimal sfvs.
Corollary
Both weighted and unweighted versions of SFVS can be solved in
O(1.6708n ) time on chordal graphs.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Proof of the theorem:
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Proof of the theorem:
Our algorithm takes as input a chordal graph G = (V , E ) and a
vertex subset S ⊆ V , and lists all maximal S-forests of G .
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Proof of the theorem:
Our algorithm takes as input a chordal graph G = (V , E ) and a
vertex subset S ⊆ V , and lists all maximal S-forests of G .
The algorithm is a recursive branching algorithm; every maximal
S-forest of G will be present at some leaf of the corresponding
branching tree.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Idea: As long as there is an undecided simplicial vertex v :
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Idea: As long as there is an undecided simplicial vertex v :
v
place v in S−forest
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
place v in sfvs
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Idea: As long as there is an undecided simplicial vertex v :
v
place v in S−forest
place v in sfvs
Problem: All simplicial vertices are already decided to be in an
S-forest.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Idea: As long as there is an undecided simplicial vertex v :
v
place v in S−forest
place v in sfvs
Problem: All simplicial vertices are already decided to be in an
S-forest.
Solution: Hide the vertices of the S-forest that are not needed for
further decisions.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Every recursive call has input (G 0 , F , U, R), Where
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Every recursive call has input (G 0 , F , U, R), Where
F : S-forest,
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Every recursive call has input (G 0 , F , U, R), Where
F : S-forest,
U: sfvs,
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Every recursive call has input (G 0 , F , U, R), Where
F : S-forest,
U: sfvs,
R: vertices of F not needed,
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Every recursive call has input (G 0 , F , U, R), Where
F : S-forest,
U: sfvs,
R: vertices of F not needed,
G 0 = G − (U ∪ R).
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Every recursive call has input (G 0 , F , U, R), Where
F : S-forest,
U: sfvs,
R: vertices of F not needed,
G 0 = G − (U ∪ R).
Definition (Measure of an instance)
The measure of an instance (G 0 , F , U, R) is the number of
undecided vertices, i.e., the vertices in G 0 − F .
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Every recursive call has input (G 0 , F , U, R), Where
F : S-forest,
U: sfvs,
R: vertices of F not needed,
G 0 = G − (U ∪ R).
Definition (Measure of an instance)
The measure of an instance (G 0 , F , U, R) is the number of
undecided vertices, i.e., the vertices in G 0 − F .
v ∈ F is not needed for further decisions if v and none of its
neighbors belong to S.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
At branching point (G 0 , F , U, R), pick a simplicial vertex v .
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
At branching point (G 0 , F , U, R), pick a simplicial vertex v .
v is undecided.
v is decided.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v is undecided.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v is undecided.
v ∈ S and N(v ) ∩ F = ∅.
d(v ) = 2
d(v ) = 3
d(v ) = t ≥ 4
Branching vector (3, 3, 3, 3)
Branching vector (4, 4, 4, 4, 2, 4)
Branching vector (1, t + 1, t + 1, · · · , t + 1), where the term
t + 1 apears t + 1 times
v ∈ S and N(v ) ∩ F 6= ∅.
Branching vector (1, t), where t ≥ 2.
v∈
/ S and N(v ) ∩ F = ∅.
d(v ) = 2
d(v ) = 3
d(v ) = t ≥ 4
Branching vector (3, 3, 3, 3)
Branching vector (4, 4, 4, 4, 2, 4)
Branching vector (1, t + 1, t + 1, · · · , t + 1), where the term
t + 1 apears t + 1 times
v∈
/ S and N(v ) ∩ F 6= ∅.
Branching vector (1, t), where t ≥ 2.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v is undecided.
v ∈ S and N(v ) ∩ F = ∅.
d(v ) = 2
d(v ) = 3
d(v ) = t ≥ 4
Branching vector (3, 3, 3, 3)
Branching vector (4, 4, 4, 4, 2, 4)
Branching vector (1, t + 1, t + 1, · · · , t + 1), where the term
t + 1 apears t + 1 times
v ∈ S and N(v ) ∩ F 6= ∅.
Branching vector (1, t), where t ≥ 2.
v∈
/ S and N(v ) ∩ F = ∅.
d(v ) = 2
d(v ) = 3
d(v ) = t ≥ 4
Branching vector (3, 3, 3, 3)
Branching vector (4, 4, 4, 4, 2, 4)
Branching vector (1, t + 1, t + 1, · · · , t + 1), where the term
t + 1 apears t + 1 times
v∈
/ S and N(v ) ∩ F 6= ∅.
Branching vector (1, t), where t ≥ 2.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
All v , u and w are undecided.
v
put v in F
put v in U
put w in F and u in U
put u in F and w in U
put u and w in U
drops by 3
drops by 3
drops by 3
u and w are in F
drops by 3
v
v is in S.
•
u
w
Vertices in F
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
All v , u and w are undecided.
v
put v in F
put v in U
put w in F and u in U
put u in F and w in U
put u and w in U
drops by 3
drops by 3
drops by 3
u and w are in F
drops by 3
v
v is in S.
•
u
w
Vertices in F
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
All v , u and w are undecided.
v
put v in F
put v in U
put w in F and u in U
put u in F and w in U
put u and w in U
drops by 3
drops by 3
drops by 3
u and w are in F
drops by 3
v
v is in S.
•
u
w
Vertices in F
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
All v , u and w are undecided.
v
put v in F
put v in U
put w in F and u in U
put u in F and w in U
put u and w in U
drops by 3
drops by 3
drops by 3
u and w are in F
drops by 3
v
v is in S.
•
u
w
Vertices in F
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
All v , u and w are undecided.
v
put v in F
put v in U
put w in F and u in U
put u in F and w in U
put u and w in U
drops by 3
drops by 3
drops by 3
u and w are in F
drops by 3
v
v is in S.
•
u
w
Vertices in F
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
All v , u and w are undecided.
v
put v in F
put v in U
put w in F and u in U
put u in F and w in U
put u and w in U
drops by 3
drops by 3
drops by 3
u and w are in F
drops by 3
v
v is in S.
•
u
w
Vertices in F
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
All v , u and w are undecided.
v
put v in F
put v in U
put w in F and u in U
put u in F and w in U
put u and w in U
drops by 3
drops by 3
drops by 3
u and w are in F
drops by 3
v
v is in S.
•
u
w
Vertices in F
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
All v , u and w are undecided.
v
put v in F
put v in U
put w in F and u in U
put u in F and w in U
put u and w in U
drops by 3
drops by 3
drops by 3
u and w are in F
drops by 3
v
v is in S.
•
u
w
Vertices in F
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
All v , u and w are undecided.
v
put v in F
put v in U
put w in F and u in U
put u in F and w in U
put u and w in U
drops by 3
drops by 3
drops by 3
u and w are in F
drops by 3
v
v is in S.
•
u
w
Vertices in F
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
All v , u and w are undecided.
v
put v in F
put v in U
put w in F and u in U
put u in F and w in U
put u and w in U
drops by 3
drops by 3
drops by 3
u and w are in F
drops by 3
v
v is in S.
•
u
w
Vertices in F
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
All v , u and w are undecided.
v
put v in F
put v in U
put w in F and u in U
put u in F and w in U
put u and w in U
drops by 3
drops by 3
drops by 3
u and w are in F
drops by 3
v
v is in S.
•
u
w
Vertices in F
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
All v , u and w are undecided.
v
put v in F
put v in U
put w in F and u in U
put u in F and w in U
put u and w in U
drops by 3
drops by 3
drops by 3
u and w are in F
drops by 3
v
v is in S.
•
u
w
Vertices in F
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
All v , u and w are undecided.
v
put v in F
put v in U
put w in F and u in U
put u in F and w in U
put u and w in U
drops by 3
drops by 3
drops by 3
u and w are in F
drops by 3
v
v is in S.
•
u
w
Vertices in F
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
All v , u and w are undecided.
v
put v in F
put v in U
put w in F and u in U
put u in F and w in U
put u and w in U
drops by 3
drops by 3
drops by 3
u and w are in F
drops by 3
v
v is in S.
•
u
w
Vertices in F
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
All v , u and w are undecided.
v
put v in F
put v in U
put w in F and u in U
put u in F and w in U
put u and w in U
drops by 3
drops by 3
drops by 3
u and w are in F
drops by 3
v
v is in S.
•
u
w
Vertices in F
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v is undecided.
v ∈ S and N(v ) ∩ F = ∅.
d(v ) = 2
d(v ) = 3
d(v ) = t ≥ 4
Branching vector (3, 3, 3, 3)
Branching vector (4, 4, 4, 4, 2, 4)
Branching vector (1, t + 1, t + 1, · · · , t + 1), where the term
t + 1 apears t + 1 times
v ∈ S and N(v ) ∩ F 6= ∅.
Branching vector (1, t), where t ≥ 2.
v∈
/ S and N(v ) ∩ F = ∅.
d(v ) = 2
d(v ) = 3
d(v ) = t ≥ 4
Branching vector (3, 3, 3, 3)
Branching vector (4, 4, 4, 4, 2, 4)
Branching vector (1, t + 1, t + 1, · · · , t + 1), where the term
t + 1 apears t + 1 times
v∈
/ S and N(v ) ∩ F 6= ∅.
Branching vector (1, t), where t ≥ 2.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v is undecided.
v ∈ S and N(v ) ∩ F = ∅.
d(v ) = 2
d(v ) = 3
d(v ) = t ≥ 4
Branching vector (3, 3, 3, 3)
Branching vector (4, 4, 4, 4, 2, 4)
Branching vector (1, t + 1, t + 1, · · · , t + 1), where the term
t + 1 apears t + 1 times
v ∈ S and N(v ) ∩ F 6= ∅.
Branching vector (1, t), where t ≥ 2.
v∈
/ S and N(v ) ∩ F = ∅.
d(v ) = 2
d(v ) = 3
d(v ) = t ≥ 4
Branching vector (3, 3, 3, 3)
Branching vector (4, 4, 4, 4, 2, 4)
Branching vector (1, t + 1, t + 1, · · · , t + 1), where the term
t + 1 apears t + 1 times
v∈
/ S and N(v ) ∩ F 6= ∅.
Branching vector (1, t), where t ≥ 2.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
All v , u, w and x are undecided.
v
put v in F
put u in F
put w in F
put v in U
put x in F
w and x are in U
u and x are in U
u and w are in U
drops by 4
drops by 4
drops by 4
put u, w and x in U
put v in
x and u are in F
drops by 4
drops by 2
v
v is in S.
u
x
w
•
Vertices in F
If at most one of w and x is in F then we could add v to F .
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
put u in U
Subset Feedback Vertex Set on chordal graphs
drops by 4
Introduction and Definitions
The Result
All v , u, w and x are undecided.
v
put v in F
put u in F
put w in F
put v in U
put x in F
w and x are in U
u and x are in U
u and w are in U
drops by 4
drops by 4
drops by 4
put u, w and x in U
put v in
x and u are in F
drops by 4
drops by 2
v
v is in S.
u
x
w
•
Vertices in F
If at most one of w and x is in F then we could add v to F .
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
put u in U
Subset Feedback Vertex Set on chordal graphs
drops by 4
Introduction and Definitions
The Result
All v , u, w and x are undecided.
v
put v in F
put u in F
put w in F
put v in U
put x in F
w and x are in U
u and x are in U
u and w are in U
drops by 4
drops by 4
drops by 4
put u, w and x in U
put v in
x and u are in F
drops by 4
drops by 2
v
v is in S.
u
x
w
•
Vertices in F
If at most one of w and x is in F then we could add v to F .
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
put u in U
Subset Feedback Vertex Set on chordal graphs
drops by 4
Introduction and Definitions
The Result
All v , u, w and x are undecided.
v
put v in F
put u in F
put w in F
put v in U
put x in F
w and x are in U
u and x are in U
u and w are in U
drops by 4
drops by 4
drops by 4
put u, w and x in U
put v in
x and u are in F
drops by 4
drops by 2
v
v is in S.
u
x
w
•
Vertices in F
If at most one of w and x is in F then we could add v to F .
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
put u in U
Subset Feedback Vertex Set on chordal graphs
drops by 4
Introduction and Definitions
The Result
All v , u, w and x are undecided.
v
put v in F
put u in F
put w in F
put v in U
put x in F
w and x are in U
u and x are in U
u and w are in U
drops by 4
drops by 4
drops by 4
put u, w and x in U
put v in
x and u are in F
drops by 4
drops by 2
v
v is in S.
u
x
w
•
Vertices in F
If at most one of w and x is in F then we could add v to F .
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
put u in U
Subset Feedback Vertex Set on chordal graphs
drops by 4
Introduction and Definitions
The Result
All v , u, w and x are undecided.
v
put v in F
put u in F
put w in F
put v in U
put x in F
w and x are in U
u and x are in U
u and w are in U
drops by 4
drops by 4
drops by 4
put u, w and x in U
put v in
x and u are in F
drops by 4
drops by 2
v
v is in S.
u
x
w
•
Vertices in F
If at most one of w and x is in F then we could add v to F .
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
put u in U
Subset Feedback Vertex Set on chordal graphs
drops by 4
Introduction and Definitions
The Result
All v , u, w and x are undecided.
v
put v in F
put u in F
put w in F
put v in U
put x in F
w and x are in U
u and x are in U
u and w are in U
drops by 4
drops by 4
drops by 4
put u, w and x in U
put v in
x and u are in F
drops by 4
drops by 2
v
v is in S.
u
x
w
•
Vertices in F
If at most one of w and x is in F then we could add v to F .
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
put u in U
Subset Feedback Vertex Set on chordal graphs
drops by 4
Introduction and Definitions
The Result
All v , u, w and x are undecided.
v
put v in F
put u in F
put w in F
put v in U
put x in F
w and x are in U
u and x are in U
u and w are in U
drops by 4
drops by 4
drops by 4
put u, w and x in U
put v in
x and u are in F
drops by 4
drops by 2
v
v is in S.
u
x
w
•
Vertices in F
If at most one of w and x is in F then we could add v to F .
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
put u in U
Subset Feedback Vertex Set on chordal graphs
drops by 4
Introduction and Definitions
The Result
All v , u, w and x are undecided.
v
put v in F
put u in F
put w in F
put v in U
put x in F
w and x are in U
u and x are in U
u and w are in U
drops by 4
drops by 4
drops by 4
put u, w and x in U
put v in
x and u are in F
drops by 4
drops by 2
v
v is in S.
u
x
w
•
Vertices in F
If at most one of w and x is in F then we could add v to F .
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
put u in U
Subset Feedback Vertex Set on chordal graphs
drops by 4
Introduction and Definitions
The Result
All v , u, w and x are undecided.
v
put v in F
put u in F
put w in F
put v in U
put x in F
w and x are in U
u and x are in U
u and w are in U
drops by 4
drops by 4
drops by 4
put u, w and x in U
put v in
x and u are in F
drops by 4
drops by 2
v
v is in S.
u
x
w
•
Vertices in F
If at most one of w and x is in F then we could add v to F .
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
put u in U
Subset Feedback Vertex Set on chordal graphs
drops by 4
Introduction and Definitions
The Result
All v , u, w and x are undecided.
v
put v in F
put u in F
put w in F
put v in U
put x in F
w and x are in U
u and x are in U
u and w are in U
drops by 4
drops by 4
drops by 4
put u, w and x in U
put v in
x and u are in F
drops by 4
drops by 2
v
v is in S.
u
x
w
•
Vertices in F
If at most one of w and x is in F then we could add v to F .
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
put u in U
Subset Feedback Vertex Set on chordal graphs
drops by 4
Introduction and Definitions
The Result
All v , u, w and x are undecided.
v
put v in F
put u in F
put w in F
put v in U
put x in F
w and x are in U
u and x are in U
u and w are in U
drops by 4
drops by 4
drops by 4
put u, w and x in U
put v in
x and u are in F
drops by 4
drops by 2
v
v is in S.
u
x
w
•
Vertices in F
If at most one of w and x is in F then we could add v to F .
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
put u in U
Subset Feedback Vertex Set on chordal graphs
drops by 4
Introduction and Definitions
The Result
All v , u, w and x are undecided.
v
put v in F
put u in F
put w in F
put v in U
put x in F
w and x are in U
u and x are in U
u and w are in U
drops by 4
drops by 4
drops by 4
put u, w and x in U
put v in
x and u are in F
drops by 4
drops by 2
v
v is in S.
u
x
w
•
Vertices in F
If at most one of w and x is in F then we could add v to F .
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
put u in U
Subset Feedback Vertex Set on chordal graphs
drops by 4
Introduction and Definitions
The Result
All v , u, w and x are undecided.
v
put v in F
put u in F
put w in F
put v in U
put x in F
w and x are in U
u and x are in U
u and w are in U
drops by 4
drops by 4
drops by 4
put u, w and x in U
put v in
x and u are in F
drops by 4
drops by 2
v
v is in S.
u
x
w
•
Vertices in F
If at most one of w and x is in F then we could add v to F .
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
put u in U
Subset Feedback Vertex Set on chordal graphs
drops by 4
Introduction and Definitions
The Result
All v , u, w and x are undecided.
v
put v in F
put u in F
put w in F
put v in U
put x in F
w and x are in U
u and x are in U
u and w are in U
drops by 4
drops by 4
drops by 4
put u, w and x in U
put v in
x and u are in F
drops by 4
drops by 2
v
v is in S.
u
x
w
•
Vertices in F
If at most one of w and x is in F then we could add v to F .
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
put u in U
Subset Feedback Vertex Set on chordal graphs
drops by 4
Introduction and Definitions
The Result
All v , u, w and x are undecided.
v
put v in F
put u in F
put w in F
put v in U
put x in F
w and x are in U
u and x are in U
u and w are in U
drops by 4
drops by 4
drops by 4
put u, w and x in U
put v in
x and u are in F
drops by 4
drops by 2
v
v is in S.
u
x
w
•
Vertices in F
If at most one of w and x is in F then we could add v to F .
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
put u in U
Subset Feedback Vertex Set on chordal graphs
drops by 4
Introduction and Definitions
The Result
All v , u, w and x are undecided.
v
put v in F
put u in F
put w in F
put v in U
put x in F
w and x are in U
u and x are in U
u and w are in U
drops by 4
drops by 4
drops by 4
put u, w and x in U
put u in F
drops by 4
drops by 2
x and u are in F
v
v is in S.
u
x
w
•
Vertices in F
If at most one of w and x is in F then we could add v to F .
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
put u in U
Subset Feedback Vertex Set on chordal graphs
drops by 4
Introduction and Definitions
The Result
All v , u, w and x are undecided.
v
put v in F
put u in F
put w in F
put v in U
put x in F
w and x are in U
u and x are in U
u and w are in U
drops by 4
drops by 4
drops by 4
put u, w and x in U
put u in F
drops by 4
drops by 2
x and u are in F
v
v is in S.
u
x
w
•
Vertices in F
If at most one of w and x is in F then we could add v to F .
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
put u in U
Subset Feedback Vertex Set on chordal graphs
drops by 4
Introduction and Definitions
The Result
All v , u, w and x are undecided.
v
put v in F
put u in F
put w in F
put v in U
put x in F
w and x are in U
u and x are in U
u and w are in U
drops by 4
drops by 4
drops by 4
put u, w and x in U
put u in F
drops by 4
drops by 2
x and u are in F
v
v is in S.
u
x
w
•
Vertices in F
If at most one of w and x is in F then we could add v to F .
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
put u in U
Subset Feedback Vertex Set on chordal graphs
drops by 4
Introduction and Definitions
The Result
All v , u, w and x are undecided.
v
put v in F
put u in F
put w in F
put v in U
put x in F
w and x are in U
u and x are in U
u and w are in U
drops by 4
drops by 4
drops by 4
put u, w and x in U
put u in F
drops by 4
drops by 2
x and u are in F
v
v is in S.
u
x
w
•
Vertices in F
If at most one of w and x is in F then we could add v to F .
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
put u in U
Subset Feedback Vertex Set on chordal graphs
drops by 4
Introduction and Definitions
The Result
All v , u, w and x are undecided.
v
put v in F
put u in F
put w in F
put v in U
put x in F
w and x are in U
u and x are in U
u and w are in U
drops by 4
drops by 4
drops by 4
put u, w and x in U
put u in F
drops by 4
drops by 2
x and u are in F
v
v is in S.
u
x
w
•
Vertices in F
If at most one of w and x is in F then we could add v to F .
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
put u in U
Subset Feedback Vertex Set on chordal graphs
drops by 4
Introduction and Definitions
The Result
All v , u, w and x are undecided.
v
put v in F
put u in F
put w in F
put v in U
put x in F
w and x are in U
u and x are in U
u and w are in U
drops by 4
drops by 4
drops by 4
put u, w and x in U
put u in F
drops by 4
drops by 2
x and u are in F
v
v is in S.
u
x
w
•
Vertices in F
If at most one of w and x is in F then we could add v to F .
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
put u in U
Subset Feedback Vertex Set on chordal graphs
drops by 4
Introduction and Definitions
The Result
All v , u, w and x are undecided.
v
put v in F
put u in F
put w in F
put v in U
put x in F
w and x are in U
u and x are in U
u and w are in U
drops by 4
drops by 4
drops by 4
put u, w and x in U
put u in F
drops by 4
drops by 2
x and u are in F
v
v is in S.
u
x
w
•
Vertices in F
If at most one of w and x is in F then we could add v to F .
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
put u in U
Subset Feedback Vertex Set on chordal graphs
drops by 4
Introduction and Definitions
The Result
All v , u, w and x are undecided.
v
put v in F
put u in F
put w in F
put v in U
put x in F
w and x are in U
u and x are in U
u and w are in U
drops by 4
drops by 4
drops by 4
put u, w and x in U
put u in F
drops by 4
drops by 2
x and u are in F
v
v is in S.
u
x
w
•
Vertices in F
If at most one of w and x is in F then we could add v to F .
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
put u in U
Subset Feedback Vertex Set on chordal graphs
drops by 4
Introduction and Definitions
The Result
v is undecided.
v ∈ S and N(v ) ∩ F = ∅.
d(v ) = 2
d(v ) = 3
d(v ) = t ≥ 4
Branching vector (3, 3, 3, 3)
Branching vector (4, 4, 4, 4, 2, 4)
Branching vector (1, t + 1, t + 1, · · · , t + 1), where the term
t + 1 apears t + 1 times
v ∈ S and N(v ) ∩ F 6= ∅.
Branching vector (1, t), where t ≥ 2.
v∈
/ S and N(v ) ∩ F = ∅.
d(v ) = 2
d(v ) = 3
d(v ) = t ≥ 4
Branching vector (3, 3, 3, 3)
Branching vector (4, 4, 4, 4, 2, 4)
Branching vector (1, t + 1, t + 1, · · · , t + 1), where the term
t + 1 apears t + 1 times
v∈
/ S and N(v ) ∩ F 6= ∅.
Branching vector (1, t), where t ≥ 2.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v is undecided.
v ∈ S and N(v ) ∩ F = ∅.
d(v ) = 2
d(v ) = 3
d(v ) = t ≥ 4
Branching vector (3, 3, 3, 3)
Branching vector (4, 4, 4, 4, 2, 4)
Branching vector (1, t + 1, t + 1, · · · , t + 1), where the term
t + 1 apears t + 1 times
v ∈ S and N(v ) ∩ F 6= ∅.
Branching vector (1, t), where t ≥ 2.
v∈
/ S and N(v ) ∩ F = ∅.
d(v ) = 2
d(v ) = 3
d(v ) = t ≥ 4
Branching vector (3, 3, 3, 3)
Branching vector (4, 4, 4, 4, 2, 4)
Branching vector (1, t + 1, t + 1, · · · , t + 1), where the term
t + 1 apears t + 1 times
v∈
/ S and N(v ) ∩ F 6= ∅.
Branching vector (1, t), where t ≥ 2.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
All v , u1 , u2 , · · · , ut are undecided.
v
put v in U
drops by 1
put v in F
put u 1 in F
put u 2 in F
. . .
put u t in F
N(v) − u 1 are in U
N(v) − u 2 are in U
. . .
N(v) − u t are in U
drops by t+1
drops by t+1
. . .
drops by t+1
put all N(v) in U
drops by t+1
v
v is in S.
•
u1
u2
...
ut
Vertices in F
Since v belongs to S it is not possible to have v and at more than
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
All v , u1 , u2 , · · · , ut are undecided.
v
put v in U
drops by 1
put v in F
put u 1 in F
put u 2 in F
. . .
put u t in F
N(v) − u 1 are in U
N(v) − u 2 are in U
. . .
N(v) − u t are in U
drops by t+1
drops by t+1
. . .
drops by t+1
put all N(v) in U
drops by t+1
v
v is in S.
•
u1
u2
...
ut
Vertices in F
Since v belongs to S it is not possible to have v and at more than
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
All v , u1 , u2 , · · · , ut are undecided.
v
put v in U
drops by 1
put v in F
put u 1 in F
put u 2 in F
. . .
put u t in F
N(v) − u 1 are in U
N(v) − u 2 are in U
. . .
N(v) − u t are in U
drops by t+1
drops by t+1
. . .
drops by t+1
put all N(v) in U
drops by t+1
v
v is in S.
•
u1
u2
...
ut
Vertices in F
Since v belongs to S it is not possible to have v and at more than
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
All v , u1 , u2 , · · · , ut are undecided.
v
put v in U
drops by 1
put v in F
put u 1 in F
put u 2 in F
. . .
put u t in F
N(v) − u 1 are in U
N(v) − u 2 are in U
. . .
N(v) − u t are in U
drops by t+1
drops by t+1
. . .
drops by t+1
put all N(v) in U
drops by t+1
v
v is in S.
•
u1
u2
...
ut
Vertices in F
Since v belongs to S it is not possible to have v and at more than
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
All v , u1 , u2 , · · · , ut are undecided.
v
put v in U
drops by 1
put v in F
put u 1 in F
put u 2 in F
. . .
put u t in F
N(v) − u 1 are in U
N(v) − u 2 are in U
. . .
N(v) − u t are in U
drops by t+1
drops by t+1
. . .
drops by t+1
put all N(v) in U
drops by t+1
v
v is in S.
•
u1
u2
...
ut
Vertices in F
Since v belongs to S it is not possible to have v and at more than
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
All v , u1 , u2 , · · · , ut are undecided.
v
put v in U
drops by 1
put v in F
put u 1 in F
put u 2 in F
. . .
put u t in F
N(v) − u 1 are in U
N(v) − u 2 are in U
. . .
N(v) − u t are in U
drops by t+1
drops by t+1
. . .
drops by t+1
put all N(v) in U
drops by t+1
v
v is in S.
•
u1
u2
...
ut
Vertices in F
Since v belongs to S it is not possible to have v and at more than
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
All v , u1 , u2 , · · · , ut are undecided.
v
put v in U
drops by 1
put v in F
put u 1 in F
put u 2 in F
. . .
put u t in F
N(v) − u 1 are in U
N(v) − u 2 are in U
. . .
N(v) − u t are in U
drops by t+1
drops by t+1
. . .
drops by t+1
put all N(v) in U
drops by t+1
v
v is in S.
•
u1
u2
...
ut
Vertices in F
Since v belongs to S it is not possible to have v and at more than
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
All v , u1 , u2 , · · · , ut are undecided.
v
put v in U
drops by 1
put v in F
put u 1 in F
put u 2 in F
. . .
put u t in F
N(v) − u 1 are in U
N(v) − u 2 are in U
. . .
N(v) − u t are in U
drops by t+1
drops by t+1
. . .
drops by t+1
put all N(v) in U
drops by t+1
v
v is in S.
•
u1
u2
...
ut
Vertices in F
Since v belongs to S it is not possible to have v and at more than
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
All v , u1 , u2 , · · · , ut are undecided.
v
put v in U
drops by 1
put v in F
put u 1 in F
put u 2 in F
. . .
put u t in F
N(v) − u 1 are in U
N(v) − u 2 are in U
. . .
N(v) − u t are in U
drops by t+1
drops by t+1
. . .
drops by t+1
put all N(v) in U
drops by t+1
v
v is in S.
•
u1
u2
...
ut
Vertices in F
Since v belongs to S it is not possible to have v and at more than
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
All v , u1 , u2 , · · · , ut are undecided.
v
put v in U
drops by 1
put v in F
put u 1 in F
put u 2 in F
. . .
put u t in F
N(v) − u 1 are in U
N(v) − u 2 are in U
. . .
N(v) − u t are in U
drops by t+1
drops by t+1
. . .
drops by t+1
put all N(v) in U
drops by t+1
v
v is in S.
•
u1
u2
...
ut
Vertices in F
Since v belongs to S it is not possible to have v and at more than
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
All v , u1 , u2 , · · · , ut are undecided.
v
put v in U
drops by 1
put v in F
put u 1 in F
put u 2 in F
. . .
put u t in F
N(v) − u 1 are in U
N(v) − u 2 are in U
. . .
N(v) − u t are in U
drops by t+1
drops by t+1
. . .
drops by t+1
put all N(v) in U
drops by t+1
v
v is in S.
•
u1
u2
...
ut
Vertices in F
Since v belongs to S it is not possible to have v and at more than
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
All v , u1 , u2 , · · · , ut are undecided.
v
put v in U
drops by 1
put v in F
put u 1 in F
put u 2 in F
. . .
put u t in F
N(v) − u 1 are in U
N(v) − u 2 are in U
. . .
N(v) − u t are in U
drops by t+1
drops by t+1
. . .
drops by t+1
put all N(v) in U
drops by t+1
v
v is in S.
•
u1
u2
...
ut
Vertices in F
Since v belongs to S it is not possible to have v and at more than
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
All v , u1 , u2 , · · · , ut are undecided.
v
put v in U
drops by 1
put v in F
put u 1 in F
put u 2 in F
. . .
put u t in F
N(v) − u 1 are in U
N(v) − u 2 are in U
. . .
N(v) − u t are in U
drops by t+1
drops by t+1
. . .
drops by t+1
put all N(v) in U
drops by t+1
v
v is in S.
•
u1
u2
...
ut
Vertices in F
Since v belongs to S it is not possible to have v and at more than
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
All v , u1 , u2 , · · · , ut are undecided.
v
put v in U
drops by 1
put v in F
put u 1 in F
put u 2 in F
. . .
put u t in F
N(v) − u 1 are in U
N(v) − u 2 are in U
. . .
N(v) − u t are in U
drops by t+1
drops by t+1
. . .
drops by t+1
put all N(v) in U
drops by t+1
v
v is in S.
•
u1
u2
...
ut
Vertices in F
Since v belongs to S it is not possible to have v and at more than
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
All v , u1 , u2 , · · · , ut are undecided.
v
put v in U
drops by 1
put v in F
put u 1 in F
put u 2 in F
. . .
put u t in F
N(v) − u 1 are in U
N(v) − u 2 are in U
. . .
N(v) − u t are in U
drops by t+1
drops by t+1
. . .
drops by t+1
put all N(v) in U
drops by t+1
v
v is in S.
•
u1
u2
...
ut
Vertices in F
Since v belongs to S it is not possible to have v and at more than
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
All v , u1 , u2 , · · · , ut are undecided.
v
put v in U
drops by 1
put v in F
put u 1 in F
put u 2 in F
. . .
put u t in F
N(v) − u 1 are in U
N(v) − u 2 are in U
. . .
N(v) − u t are in U
drops by t+1
drops by t+1
. . .
drops by t+1
put all N(v) in U
drops by t+1
v
v is in S.
•
u1
u2
...
ut
Vertices in F
Since v belongs to S it is not possible to have v and at more than
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
All v , u1 , u2 , · · · , ut are undecided.
v
put v in U
drops by 1
put v in F
put u 1 in F
put u 2 in F
. . .
put u t in F
N(v) − u 1 are in U
N(v) − u 2 are in U
. . .
N(v) − u t are in U
drops by t+1
drops by t+1
. . .
drops by t+1
put all N(v) in U
drops by t+1
v
v is in S.
•
u1
u2
...
ut
Vertices in F
Since v belongs to S it is not possible to have v and at more than
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
All v , u1 , u2 , · · · , ut are undecided.
v
put v in U
drops by 1
put v in F
put u 1 in F
put u 2 in F
. . .
put u t in F
N(v) − u 1 are in U
N(v) − u 2 are in U
. . .
N(v) − u t are in U
drops by t+1
drops by t+1
. . .
drops by t+1
put all N(v) in U
drops by t+1
v
v is in S.
•
u1
u2
...
ut
Vertices in F
Since v belongs to S it is not possible to have v and at more than
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
All v , u1 , u2 , · · · , ut are undecided.
v
put v in U
drops by 1
put v in F
put u 1 in F
put u 2 in F
. . .
put u t in F
N(v) − u 1 are in U
N(v) − u 2 are in U
. . .
N(v) − u t are in U
drops by t+1
drops by t+1
. . .
drops by t+1
put all N(v) in U
drops by t+1
v
v is in S.
•
u1
u2
...
ut
Vertices in F
Since v belongs to S it is not possible to have v and at more than
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v is undecided.
v ∈ S and N(v ) ∩ F = ∅.
d(v ) = 2
d(v ) = 3
d(v ) = t ≥ 4
Branching vector (3, 3, 3, 3)
Branching vector (4, 4, 4, 4, 2, 4)
Branching vector (1, t + 1, t + 1, · · · , t + 1), where the term
t + 1 apears t + 1 times
v ∈ S and N(v ) ∩ F 6= ∅.
Branching vector (1, t), where t ≥ 2.
v∈
/ S and N(v ) ∩ F = ∅.
d(v ) = 2
d(v ) = 3
d(v ) = t ≥ 4
Branching vector (3, 3, 3, 3)
Branching vector (4, 4, 4, 4, 2, 4)
Branching vector (1, t + 1, t + 1, · · · , t + 1), where the term
t + 1 apears t + 1 times
v∈
/ S and N(v ) ∩ F 6= ∅.
Branching vector (1, t), where t ≥ 2.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v is undecided but some of its neighbors are decided.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v is undecided but some of its neighbors are decided.
In the case that |N(v ) ∩ F | ≥ 2, v can not be added to F .
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v is undecided but some of its neighbors are decided.
In the case that |N(v ) ∩ F | ≥ 2, v can not be added to F .
v
u1
u2
...
ut
v belongs to S.
Then add v to U. The measure drops by one without any
branching.
So, exactly one of its neighbors belongs to F .
Without loss of generality assume u1 is in F .
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v is undecided but some of its neighbors are decided.
In the case that |N(v ) ∩ F | ≥ 2, v can not be added to F .
v
u1
u2
...
ut
v belongs to S.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v is undecided but some of its neighbors are decided.
In the case that |N(v ) ∩ F | ≥ 2, v can not be added to F .
v
u1
u2
...
ut
v belongs to S.
Then add v to U. The measure drops by one without any
branching.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v is undecided but some of its neighbors are decided.
In the case that |N(v ) ∩ F | ≥ 2, v can not be added to F .
v
u1
u2
...
ut
v belongs to S.
Then add v to U. The measure drops by one without any
branching.
So, exactly one of its neighbors belongs to F .
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v is undecided but some of its neighbors are decided.
In the case that |N(v ) ∩ F | ≥ 2, v can not be added to F .
v
u1
u2
...
ut
v belongs to S.
Then add v to U. The measure drops by one without any
branching.
So, exactly one of its neighbors belongs to F .
Without loss of generality assume u1 is in F .
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v ∈ S.
u1 is decided.
N[v ] − u1 are undecided.
v
put v in U
put v in F
N(v) − u1 are in U
drops by 1
drops by t
v
v is in S.
•
u1
u2
...
ut
Vertices in F
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v ∈ S.
u1 is decided.
N[v ] − u1 are undecided.
v
put v in U
put v in F
N(v) − u1 are in U
drops by 1
drops by t
v
v is in S.
•
u1
u2
...
ut
Vertices in F
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v ∈ S.
u1 is decided.
N[v ] − u1 are undecided.
v
put v in U
put v in F
N(v) − u1 are in U
drops by 1
drops by t
v
v is in S.
•
u1
u2
...
ut
Vertices in F
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v ∈ S.
u1 is decided.
N[v ] − u1 are undecided.
v
put v in U
put v in F
N(v) − u1 are in U
drops by 1
drops by t
v
v is in S.
•
u1
u2
...
ut
Vertices in F
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v ∈ S.
u1 is decided.
N[v ] − u1 are undecided.
v
put v in U
put v in F
N(v) − u1 are in U
drops by 1
drops by t
v
v is in S.
•
u1
u2
...
ut
Vertices in F
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v ∈ S.
u1 is decided.
N[v ] − u1 are undecided.
v
put v in U
put v in F
N(v) − u1 are in U
drops by 1
drops by t
v
v is in S.
•
u1
u2
...
ut
Vertices in F
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v ∈ S.
u1 is decided.
N[v ] − u1 are undecided.
v
put v in U
put v in F
N(v) − u1 are in U
drops by 1
drops by t
v
v is in S.
•
u1
u2
...
ut
Vertices in F
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v ∈ S.
u1 is decided.
N[v ] − u1 are undecided.
v
put v in U
put v in F
N(v) − u1 are in U
drops by 1
drops by t
v
v is in S.
•
u1
u2
...
ut
Vertices in F
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v ∈ S.
u1 is decided.
N[v ] − u1 are undecided.
v
put v in U
put v in F
N(v) − u1 are in U
drops by 1
drops by t
v
v is in S.
•
u1
u2
...
ut
Vertices in F
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v is undecided.
v ∈ S and N(v ) ∩ F = ∅.
d(v ) = 2
d(v ) = 3
d(v ) = t ≥ 4
Branching vector (3, 3, 3, 3)
Branching vector (4, 4, 4, 4, 2, 4)
Branching vector (1, t + 1, t + 1, · · · , t + 1), where the term
t + 1 apears t + 1 times
v ∈ S and N(v ) ∩ F 6= ∅.
Branching vector (1, t), where t ≥ 2.
v∈
/ S and N(v ) ∩ F = ∅.
d(v ) = 2
d(v ) = 3
d(v ) = t ≥ 4
Branching vector (3, 3, 3, 3)
Branching vector (4, 4, 4, 4, 2, 4)
Branching vector (1, t + 1, t + 1, · · · , t + 1), where the term
t + 1 apears t + 1 times
v∈
/ S and N(v ) ∩ F 6= ∅.
Branching vector (1, t), where t ≥ 2.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v is undecided.
v ∈ S and N(v ) ∩ F = ∅.
d(v ) = 2
d(v ) = 3
d(v ) = t ≥ 4
Branching vector (3, 3, 3, 3)
Branching vector (4, 4, 4, 4, 2, 4)
Branching vector (1, t + 1, t + 1, · · · , t + 1), where the term
t + 1 apears t + 1 times
v ∈ S and N(v ) ∩ F 6= ∅.
Branching vector (1, t), where t ≥ 2.
v∈
/ S and N(v ) ∩ F = ∅.
d(v ) = 2
d(v ) = 3
d(v ) = t ≥ 4
Branching vector (3, 3, 3, 3)
Branching vector (4, 4, 4, 4, 2, 4)
Branching vector (1, t + 1, t + 1, · · · , t + 1), where the term
t + 1 apears t + 1 times
v∈
/ S and N(v ) ∩ F 6= ∅.
Branching vector (1, t), where t ≥ 2.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v is decided.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v is decided.
Since v belongs to G 0 and decided then it belongs to F .
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v is decided.
Since v belongs to G 0 and decided then it belongs to F .
v ∈ S.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v is decided.
Since v belongs to G 0 and decided then it belongs to F .
v ∈ S.
d(v ) = 2
d(v ) = t ≥ 3
Branching vector (1, 2)
Branching vector (t, t, t, · · · , t), where the term
t apears t + 1 times
v∈
/ S.
Branching vector (1, t), where t ≥ 2.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v is decided.
Since v belongs to G 0 and decided then it belongs to F .
v ∈ S.
d(v ) = 2
d(v ) = t ≥ 3
Branching vector (1, 2)
Branching vector (t, t, t, · · · , t), where the term
t apears t + 1 times
v∈
/ S.
Branching vector (1, t), where t ≥ 2.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v ∈ S and decided.
Both u and w cannot be decided since, it gives an S-cycle.
If one of u or w is decided then we should put the other one
in U and no need to branch.
v
u
w
Hence, assume both u and w are undecided.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v ∈ S and decided.
Both u and w cannot be decided since, it gives an S-cycle.
If one of u or w is decided then we should put the other one
in U and no need to branch.
v
u
w
Hence, assume both u and w are undecided.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v ∈ S and decided.
Both u and w cannot be decided since, it gives an S-cycle.
If one of u or w is decided then we should put the other one
in U and no need to branch.
v
u
w
Hence, assume both u and w are undecided.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v ∈ S and decided.
Both u and w cannot be decided since, it gives an S-cycle.
If one of u or w is decided then we should put the other one
in U and no need to branch.
v
u
w
Hence, assume both u and w are undecided.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
u and w are undecided.
v ∈ S and is decided.
v
put u in F
put u in U
w is in U
drops by 2
drops by 1
v
u
•
w
Vertices in F
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
u and w are undecided.
v ∈ S and is decided.
v
put u in F
put u in U
w is in U
drops by 2
drops by 1
v
u
•
w
Vertices in F
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
u and w are undecided.
v ∈ S and is decided.
v
put u in F
put u in U
w is in U
drops by 2
drops by 1
v
u
•
w
Vertices in F
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
u and w are undecided.
v ∈ S and is decided.
v
put u in F
put u in U
w is in U
drops by 2
drops by 1
v
u
•
w
Vertices in F
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
u and w are undecided.
v ∈ S and is decided.
v
put u in F
put u in U
w is in U
drops by 2
drops by 1
v
u
•
w
Vertices in F
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
u and w are undecided.
v ∈ S and is decided.
v
put u in F
put u in U
w is in U
drops by 2
drops by 1
v
u
•
w
Vertices in F
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
u and w are undecided.
v ∈ S and is decided.
v
put u in F
put u in U
w is in U
drops by 2
drops by 1
v
u
•
w
Vertices in F
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
u and w are undecided.
v ∈ S and is decided.
v
put u in F
put u in U
w is in U
drops by 2
drops by 1
v
u
•
w
Vertices in F
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
u and w are undecided.
v ∈ S and is decided.
v
put u in F
put u in U
w is in U
drops by 2
drops by 1
v
u
•
w
Vertices in F
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v is decided.
Since v belongs to G 0 and decided then it belongs to F .
v ∈ S.
d(v ) = 2
d(v ) = t ≥ 3
Branching vector (1, 2)
Branching vector (t, t, t, · · · , t), where the term
t apears t + 1 times
v∈
/ S.
Branching vector (1, t), where t ≥ 2.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v is decided.
Since v belongs to G 0 and decided then it belongs to F .
v ∈ S.
d(v ) = 2
d(v ) = t ≥ 3
Branching vector (1, 2)
Branching vector (t, t, t, · · · , t), where the term
t apears t + 1 times
v∈
/ S.
Branching vector (1, t), where t ≥ 2.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
v is decided.
Since v belongs to G 0 and decided then it belongs to F .
v ∈ S.
d(v ) = 2
d(v ) = t ≥ 3
Branching vector (1, 2)
Branching vector (t, t, t, · · · , t), where the term
t apears t + 1 times
v∈
/ S.
Branching vector (1, t), where t ≥ 2.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Number of leaves of the search tree:
(3, 3, 3, 3): ≈ 1.5875.
(4, 4, 4, 4, 2, 4): ≈ 1.6708.
(1, 5, 5, 5, 5, 5): ≈ 1.6595.
(1, 2): ≈ 1.6181.
(3, 3, 3): ≈ 1.4423.
Maximum number of maximal S-forests (minimal sfvs): 1.6708n
Running time of the algorithm: O(1.6708n )
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Number of leaves of the search tree:
(3, 3, 3, 3): ≈ 1.5875.
(4, 4, 4, 4, 2, 4): ≈ 1.6708.
(1, 5, 5, 5, 5, 5): ≈ 1.6595.
(1, 2): ≈ 1.6181.
(3, 3, 3): ≈ 1.4423.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Number of leaves of the search tree:
(3, 3, 3, 3): ≈ 1.5875.
(4, 4, 4, 4, 2, 4): ≈ 1.6708.
(1, 5, 5, 5, 5, 5): ≈ 1.6595.
(1, 2): ≈ 1.6181.
(3, 3, 3): ≈ 1.4423.
Maximum number of maximal S-forests (minimal sfvs): 1.6708n
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Number of leaves of the search tree:
(3, 3, 3, 3): ≈ 1.5875.
(4, 4, 4, 4, 2, 4): ≈ 1.6708.
(1, 5, 5, 5, 5, 5): ≈ 1.6595.
(1, 2): ≈ 1.6181.
(3, 3, 3): ≈ 1.4423.
Maximum number of maximal S-forests (minimal sfvs): 1.6708n
Running time of the algorithm: O(1.6708n )
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Theorem
All minimal sfvs of a chordal graph on n vertices can be listed in
O(1.6708n ) time.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Open problems:
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Open problems:
Can the upper bound 1.6708n for maximum number of
minimal sfvs be improved on chordal graphs?
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Open problems:
Can the upper bound 1.6708n for maximum number of
minimal sfvs be improved on chordal graphs?
(Recall: Lower bound: 1.5848n ∼ 10n/5 )
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Open problems:
Can the upper bound 1.6708n for maximum number of
minimal sfvs be improved on chordal graphs?
(Recall: Lower bound: 1.5848n ∼ 10n/5 )
Can the upper bound 1.6708n for maximum number of
minimal sfvs be improved even on split graphs?
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Open problems:
Can the upper bound 1.6708n for maximum number of
minimal sfvs be improved on chordal graphs?
(Recall: Lower bound: 1.5848n ∼ 10n/5 )
Can the upper bound 1.6708n for maximum number of
minimal sfvs be improved even on split graphs?
(Lower bound: 1.4422n ∼ 3n/3 )
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Open problems:
Can the upper bound 1.6708n for maximum number of
minimal sfvs be improved on chordal graphs?
(Recall: Lower bound: 1.5848n ∼ 10n/5 )
Can the upper bound 1.6708n for maximum number of
minimal sfvs be improved even on split graphs?
(Lower bound: 1.4422n ∼ 3n/3 )
Find a faster algorithm than O(1.6708n ) for SFVS on chordal
graphs or split graphs.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Thank you
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs
Introduction and Definitions
The Result
Running time:
(3, 3, 3, 3): the branching number is ≈ 1.5875.
(4, 4, 4, 4, 2, 4): the branching number is ≈ 1.6708.
(1, t, t, t, t, . . . , t), where the term t appears t times, and
t ≥ 5: (1, 5, 5, 5, 5, 5) gives the maximum branching number
for this vector, which is ≈ 1.6595.
(1, t), t ≥ 2: (1, 2) gives the maximum branching number for
this branching vector, which is ≈ 1.6181.
(t, . . . , t), where the term t is repeated t times, and t ≥ 2:
(3, 3, 3) gives the maximum branching number for this vector,
which is ≈ 1.4423.
(t, t, . . . , t), where the term t is repeated t + 1 times, and
t ≥ 3: (3, 3, 3, 3) gives the maximum branching number for
this vector, which is ≈ 1.5875.
(1, 2): the branching number is ≈ 1.6181.
P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei
Subset Feedback Vertex Set on chordal graphs