Paths, trees and asteroids
Benjamin Lévêque
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Chordal graphs
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Chordal graphs
Definition
Graphs with no holes
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Chordal graphs
Definition
Graphs with no holes
Theorem (Gavril - 1974)
Chordal ⇐⇒ intersection graph of a family of subtrees of a tree
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Chordal graphs
Definition
Graphs with no holes
Theorem (Gavril - 1974)
Chordal ⇐⇒ intersection graph of a family of subtrees of a tree
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Chordal graphs
Definition
Graphs with no holes
Theorem (Gavril - 1974)
Chordal ⇐⇒ intersection graph of a family of subtrees of a tree
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Chordal graphs
Definition
Graphs with no holes
Theorem (Gavril - 1974)
Chordal ⇐⇒ intersection graph of a family of subtrees of a tree
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Interval graphs
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Interval graphs
Definition
Intersection graph of a family of subpaths of a path
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Interval graphs
Definition
Intersection graph of a family of subpaths of a path
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Interval graphs
Definition
Intersection graph of a family of subpaths of a path
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Interval graphs
Definition
Intersection graph of a family of subpaths of a path
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Interval graphs
Definition
Intersection graph of a family of subpaths of a path
Asteroidal triple
3 non adjacent vertices such that between any two of them, there
exists a path that does not intersect the neighborhood of the third.
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Interval graphs
Definition
Intersection graph of a family of subpaths of a path
Asteroidal triple
3 non adjacent vertices such that between any two of them, there
exists a path that does not intersect the neighborhood of the third.
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Theorem (Lekkerkerker and Boland - 1962)
Interval graph ⇐⇒ chordal + no asteroidal triple
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Theorem (Lekkerkerker and Boland - 1962)
Interval graph ⇐⇒ chordal + no asteroidal triple
⇐⇒ it does not contain :
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Path graphs
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Path graphs
Definition
Intersection graph of a family of subpaths of a tree
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Path graphs
Definition
Intersection graph of a family of subpaths of a tree
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Path graphs
Definition
Intersection graph of a family of subpaths of a tree
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Path graphs
Definition
Intersection graph of a family of subpaths of a tree
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Path graphs
Definition
Intersection graph of a family of subpaths of a tree
Question (Renz - 1970) :
Characterization by forbidden induced subgraphs ?
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Neighborhoods in path graphs
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Neighborhoods in path graphs
Remark
The neighborhood of every vertex is an interval graph
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Neighborhoods in path graphs
Remark
The neighborhood of every vertex is an interval graph
=⇒ a path graph does not contain :
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Neighborhoods in path graphs
Remark
The neighborhood of every vertex is an interval graph
=⇒ a path graph does not contain :
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Neighborhoods in path graphs
Remark
The neighborhood of every vertex is an interval graph
=⇒ a path graph does not contain :
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Neighborhoods in path graphs
Remark
The neighborhood of every vertex is an interval graph
=⇒ a path graph does not contain :
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Neighborhoods in path graphs
Remark
The neighborhood of every vertex is an interval graph
=⇒ a path graph does not contain :
Question : Is that enough ?
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Neighborhoods in path graphs
Remark
The neighborhood of every vertex is an interval graph
=⇒ a path graph does not contain :
Question : Is that enough ?
(Renz - 1970)
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Theorem (Lévêque, Maffray and Preissmann - 2009)
Path graph ⇐⇒ it does not contain :
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Theorem (Lévêque, Maffray and Preissmann - 2009)
Path graph ⇐⇒ it does not contain :
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Simplicial vertices
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Simplicial vertices
Theorem (Dirac - 1961)
Chordal, not a clique =⇒ 2 non adjacent simplicial vertices
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Simplicial vertices
Theorem (Dirac - 1961)
Chordal, not a clique =⇒ 2 non adjacent simplicial vertices
Generalization (Lévêque, Maffray and Preissmann - 2009)
Chordal, not a clique =⇒ 2 non adjacent "special" simplicial vertices
(N [v] ∩ N (V \N [v]) is a maximal separator)
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Simplicial vertices
Theorem (Dirac - 1961)
Chordal, not a clique =⇒ 2 non adjacent simplicial vertices
Generalization (Lévêque, Maffray and Preissmann - 2009)
Chordal, not a clique =⇒ 2 non adjacent "special" simplicial vertices
(N [v] ∩ N (V \N [v]) is a maximal separator)
Remark
L EX BFS and MCS do not necessarly find "special" simplicial vertices
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Simplicial vertices
Theorem (Dirac - 1961)
Chordal, not a clique =⇒ 2 non adjacent simplicial vertices
Generalization (Lévêque, Maffray and Preissmann - 2009)
Chordal, not a clique =⇒ 2 non adjacent "special" simplicial vertices
(N [v] ∩ N (V \N [v]) is a maximal separator)
Remark
L EX BFS and MCS do not necessarly find "special" simplicial vertices
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Simplicial vertices
Theorem (Dirac - 1961)
Chordal, not a clique =⇒ 2 non adjacent simplicial vertices
Generalization (Lévêque, Maffray and Preissmann - 2009)
Chordal, not a clique =⇒ 2 non adjacent "special" simplicial vertices
(N [v] ∩ N (V \N [v]) is a maximal separator)
Remark
L EX BFS and MCS do not necessarly find "special" simplicial vertices
Question : "special" simplicial elimination ordering in linear time ?
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Proof
Suppose there exists a graph minimally not path graph not in the list
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Proof
Suppose there exists a graph minimally not path graph not in the list
Q
u
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Proof
Suppose there exists a graph minimally not path graph not in the list
Q
u
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Proof
Suppose there exists a graph minimally not path graph not in the list
Q
u
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Proof
Suppose there exists a graph minimally not path graph not in the list
P
Q
u
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Proof
Suppose there exists a graph minimally not path graph not in the list
P
Q
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Proof
Suppose there exists a graph minimally not path graph not in the list
P
x
Q
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Proof
Suppose there exists a graph minimally not path graph not in the list
P
x
Q
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Proof
Suppose there exists a graph minimally not path graph not in the list
P
x
v
Q
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Proof
Suppose there exists a graph minimally not path graph not in the list
P
x
v
w
Q
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Proof
Suppose there exists a graph minimally not path graph not in the list
P
x
v
w
Q
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Proof
Suppose there exists a graph minimally not path graph not in the list
P
x
v
w
Q
u
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Proof
Suppose there exists a graph minimally not path graph not in the list
P
x
v
w
Q
u
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Proof
Suppose there exists a graph minimally not path graph not in the list
P
x
v
w
Q
u
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Proof
Suppose there exists a graph minimally not path graph not in the list
P
x
v
w
Q
u
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Directed path graphs
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Directed path graphs
Definition
Intersection graph of a family of directed subpaths of a directed tree
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Directed path graphs
Definition
Intersection graph of a family of directed subpaths of a directed tree
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Directed path graphs
Definition
Intersection graph of a family of directed subpaths of a directed tree
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Directed path graphs
Definition
Intersection graph of a family of directed subpaths of a directed tree
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Directed path graphs
Definition
Intersection graph of a family of directed subpaths of a directed tree
Theorem (Panda - 1999)
Directed path graph ⇐⇒ it does not contain :
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Directed path graphs
Definition
Intersection graph of a family of directed subpaths of a directed tree
Theorem (Panda - 1999)
Directed path graph ⇐⇒ it does not contain :
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Strong path
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Strong path
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Strong path
Property
In any directed path graph model the path between two vertices that
are linked by a strong path is directed.
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Strong path
Property
In any directed path graph model the path between two vertices that
are linked by a strong path is directed.
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Strong path
Property
In any directed path graph model the path between two vertices that
are linked by a strong path is directed.
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Strong asteroidal triple
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Strong asteroidal triple
Definition
An asteroidal triple such that between any two of the vertices of the
triple, there exists a strong path between them.
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Strong asteroidal triple
Definition
An asteroidal triple such that between any two of the vertices of the
triple, there exists a strong path between them.
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Strong asteroidal triple
Definition
An asteroidal triple such that between any two of the vertices of the
triple, there exists a strong path between them.
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Strong asteroidal triple
Definition
An asteroidal triple such that between any two of the vertices of the
triple, there exists a strong path between them.
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Strong asteroidal triple
Definition
An asteroidal triple such that between any two of the vertices of the
triple, there exists a strong path between them.
Theorem (Cameron, Hoàng, Lévêque - 2009)
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Strong asteroidal triple
Definition
An asteroidal triple such that between any two of the vertices of the
triple, there exists a strong path between them.
Theorem (Cameron, Hoàng, Lévêque - 2009)
Directed path graph ⇐⇒ chordal + no strong asteroidal triple
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Rooted path graphs
Definition
Intersection graph of a family of directed subpaths of a rooted tree
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Rooted path graphs
Definition
Intersection graph of a family of directed subpaths of a rooted tree
Open problem
Forbidden induced subgraph characterization ?
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Rooted path graphs
Definition
Intersection graph of a family of directed subpaths of a rooted tree
Open problem
Forbidden induced subgraph characterization ?
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Rooted path graphs
Definition
Intersection graph of a family of directed subpaths of a rooted tree
Open problem
Forbidden induced subgraph characterization ?
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Rooted path graphs
Definition
Intersection graph of a family of directed subpaths of a rooted tree
Open problem
Forbidden induced subgraph characterization ?
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Rooted path graphs
Definition
Intersection graph of a family of directed subpaths of a rooted tree
Open problem
Forbidden induced subgraph characterization ?
Conjecture
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Rooted path graphs
Definition
Intersection graph of a family of directed subpaths of a rooted tree
Open problem
Forbidden induced subgraph characterization ?
Conjecture
Rooted path graph ⇐⇒
chordal + no strong asteroidal triple + no weak asteroidal quadruple
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Summary of results and open problems
path graphs
directed path graphs
rooted path graphs
interval graphs
forbidden subgraphs
[LMP09]
[Pan99]
?
[LB62]
forbidden asteroids
?
[CHL09]
?
[LB62]
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Recognition algorithms
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Recognition algorithms
• Chordal graphs : O(n + m) (Rose, Tarjan, Lueker - 1976)
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Recognition algorithms
• Chordal graphs : O(n + m) (Rose, Tarjan, Lueker - 1976)
• Interval graphs : O(n + m) (Booth, Lueker - 1976) ... and many others
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Recognition algorithms
• Chordal graphs : O(n + m) (Rose, Tarjan, Lueker - 1976)
• Interval graphs : O(n + m) (Booth, Lueker - 1976) ... and many others
• Path graphs :
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Recognition algorithms
• Chordal graphs : O(n + m) (Rose, Tarjan, Lueker - 1976)
• Interval graphs : O(n + m) (Booth, Lueker - 1976) ... and many others
• Path graphs : O(n4 ) (Gavril - 1978)
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Recognition algorithms
• Chordal graphs : O(n + m) (Rose, Tarjan, Lueker - 1976)
• Interval graphs : O(n + m) (Booth, Lueker - 1976) ... and many others
• Path graphs : O(n4 ) (Gavril - 1978)
O(nm) (Schäffer - 1993)
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Recognition algorithms
• Chordal graphs : O(n + m) (Rose, Tarjan, Lueker - 1976)
• Interval graphs : O(n + m) (Booth, Lueker - 1976) ... and many others
• Path graphs : O(n4 ) (Gavril - 1978)
O(nm) (Schäffer - 1993)
O(nm) (Chaplick - 2008)
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Recognition algorithms
• Chordal graphs : O(n + m) (Rose, Tarjan, Lueker - 1976)
• Interval graphs : O(n + m) (Booth, Lueker - 1976) ... and many others
• Path graphs : O(n4 ) (Gavril - 1978)
O(nm) (Schäffer - 1993)
O(nm) (Chaplick - 2008)
O(n + m) (Dalhaus, Bailey - 1996) ??? conference paper
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Recognition algorithms
• Chordal graphs : O(n + m) (Rose, Tarjan, Lueker - 1976)
• Interval graphs : O(n + m) (Booth, Lueker - 1976) ... and many others
• Path graphs : O(n4 ) (Gavril - 1978)
O(nm) (Schäffer - 1993)
O(nm) (Chaplick - 2008)
O(n + m) (Dalhaus, Bailey - 1996) ??? conference paper
• Directed path graphs :
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Recognition algorithms
• Chordal graphs : O(n + m) (Rose, Tarjan, Lueker - 1976)
• Interval graphs : O(n + m) (Booth, Lueker - 1976) ... and many others
• Path graphs : O(n4 ) (Gavril - 1978)
O(nm) (Schäffer - 1993)
O(nm) (Chaplick - 2008)
O(n + m) (Dalhaus, Bailey - 1996) ??? conference paper
• Directed path graphs ⇐⇒ Path graph with no odd suns
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Recognition algorithms
• Chordal graphs : O(n + m) (Rose, Tarjan, Lueker - 1976)
• Interval graphs : O(n + m) (Booth, Lueker - 1976) ... and many others
• Path graphs : O(n4 ) (Gavril - 1978)
O(nm) (Schäffer - 1993)
O(nm) (Chaplick - 2008)
O(n + m) (Dalhaus, Bailey - 1996) ??? conference paper
• Directed path graphs ⇐⇒ Path graph with no odd suns
Corollary
As fast as path graph !
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Recognition algorithms
• Chordal graphs : O(n + m) (Rose, Tarjan, Lueker - 1976)
• Interval graphs : O(n + m) (Booth, Lueker - 1976) ... and many others
• Path graphs : O(n4 ) (Gavril - 1978)
O(nm) (Schäffer - 1993)
O(nm) (Chaplick - 2008)
O(n + m) (Dalhaus, Bailey - 1996) ??? conference paper
• Directed path graphs ⇐⇒ Path graph with no odd suns
Corollary
As fast as path graph !
• Rooted path graphs : O(n + m) (Dietz - 1984) ??? thesis
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