Independent Sets in Graph Classes
T. Karthick
Indian Statistical Institute Chennai Centre
Chennai - 600 113
[email protected]
Some Notations
Graph G(V, E) := Simple, Finite and Undirected.
n := |V (G)| and m := |E(G)|.
Pt := Chordless Path on t vertices.
Ct := Chordless Cycle on t vertices.
Kp,q := Complete Bipartite Graph.
Independent Sets in Graphs
Independent Set:
A set of vertices that are pairwise non-adjacent in G.
Independent Sets in Graphs
Independent Set:
A set of vertices that are pairwise non-adjacent in G.
Example
{a}, {b, d}, {a, c, e} are independent sets.
{c, e, f } is not an independent set.
Independent Sets in Graphs
Independent Set:
A set of vertices that are pairwise non-adjacent in G.
Independence Number, α(G):
Size of a maximum independent set.
Independent Sets in Graphs
Independent Set:
A set of vertices that are pairwise non-adjacent in G.
Independence Number, α(G):
Size of a maximum independent set.
Example
α(G) = 3
Maximum Weight Independent Sets
Let G be a weighted graph with weight function w on V (G).
Weighted Independence Number, αw (G):
Maximum total weight among all independent sets in G.
Maximum Weight Independent Sets
Let G be a weighted graph with weight function w on V (G).
Weighted Independence Number, αw (G):
Maximum total weight among all independent sets in G.
Example
αw (G) = 9
M AXIMUM I NDEPENDENT S ET (MIS) Problem
I NSTANCE: Graph G, a positive integer k.
Q UESTION: Does there exists an independent set I of G
such that |I| ≥ k?
M AXIMUM W EIGHT I NDEPENDENT S ET (MWIS) Problem
If S ⊆ V (G), then w(S):= Total weight of vertices in S.
I NSTANCE: Weighted graph (G, w), a positive integer k.
Q UESTION: Does there exists an independent set I of G
such that w(I) ≥ k?
MWIS reduces to MIS if w(v) = 1 for all v ∈ V (G).
? MWIS is N P -complete in general [27].
Graphs defined by forbidden induced subgraphs
Let F = {H1, H2, H3, . . .} be a family of graphs.
A graph G is said to be F-free if no induced subgraph of G is
isomorphic to Hi, for every i.
Examples
Bipartite graphs : C2k+1-free, k ≥ 1 (König’s Theorem).
Chordal graphs1 : Ck -free, k ≥ 4.
Line graphs2 : A family of 9 forbidden subgraphs (Beineke).
1
2
Every cycle of length ≥ 4 contains a chord.
Graph L(G) obtained from G, where V (L(G)) = E(G), and two vertices in L(G) are adjacent iff the
corresponding edges in G are adjacent.
Graphs defined by forbidden induced subgraphs
Let F = {H1, H2, H3, . . .} be a family of graphs.
A graph G is said to be F-free if no induced subgraph of G is
isomorphic to Hi, for every i.
Examples
Bipartite graphs : C2k+1-free, k ≥ 1 (König’s Theorem).
Chordal graphs : Ck -free, k ≥ 4.
Line graphs : A family of 9 forbidden subgraphs (Beineke).
c
Perfect graphs 3 : {C2k+1, C2k+1
}-free, k ≥ 2 (SPGT [14]).
3
Graph G with χ(H) = ω(H), ∀ H v G.
MWIS vs Graph Classes
• MWIS remains N P -complete on restricted classes of graphs
such as
– K1,4-free graphs [31].
– (K1,4, diamond)-free graphs [16].
– triangle-free graphs [36].
– Cubic or planar graphs [35].
– Graphs not containing cycles of certain length [35].
MWIS vs Graph Classes
• MWIS remains N P -complete on restricted classes of graphs
such as
– K1,4-free graphs [31].
– (K1,4, diamond)-free graphs [16].
– triangle-free graphs [36].
– Cubic or planar graphs [35].
– Graphs not containing cycles of certain length [35].
Alekseev [1]: MWIS remains N P -complete on H-free graphs,
whenever H is connected, but neither a path nor a
subdivision of the claw (K1,3).
MWIS vs H-free Graphs, H is a path
• Corneil et al. [17]: MWIS on P4-free graphs (or cographs) can
be solved in linear time.
MWIS vs H-free Graphs, H is a path
• Corneil et al. [17]: MWIS on P4-free graphs (or cographs) can
be solved in linear time.
• Lokshantov et al. [28]: MWIS on P5-free graphs can be solved
in time O(n12m) via minimal triangulations.
MWIS vs H-free Graphs, H is a path
• Corneil et al. [17]: MWIS on P4-free graphs (or cographs) can
be solved in linear time.
• Lokshantov et al. [28]: MWIS on P5-free graphs can be solved
in time O(n12m) via minimal triangulations.
• The complexity of MWIS is unknown for P6-free graphs.
MWIS vs H-free Graphs, H is a path
• Corneil et al. [17]: MWIS on P4-free graphs (or cographs) can
be solved in linear time.
• Lokshantov et al. [28]: MWIS on P5-free graphs can be solved
in time O(n12m) via minimal triangulations.
• The complexity of MWIS is unknown for P6-free graphs, and is
unknown even for (P6, C6, C5)-free graphs.
MWIS for H-free Graphs
In this talk, using clique separator decomposition techniques, we
prove the following:
MWIS can be solved in polynomial time for a subclass of P6-free
graphs.
In particular, we prove the following:
MWIS can be solved in time O(n3m) for
(P6,
4
)-free graphs4
T. Karthick, Maximum weight independent sets in (P6 , banner)-free graphs, Submitted for publication.
Clique Separators
A clique is a set of mutually adjacent vertices in G.
{a, b, c, d}, {b, d, e}, {a, b, c}, {e, f } are cliques.
But, {b, c, d, e} is not a clique.
Clique Separators
A clique is a set of mutually adjacent vertices in G.
Clique Separator: Given a connected graph G. Let Q ⊆ V (G).
Q is a clique separator if
(i) Q is a clique.
(ii) [V (G) \ Q] is disconnected.
Clique Separators
A clique is a set of mutually adjacent vertices in G.
Clique Separator: Given a connected graph G. Q ⊆ V (G).
Q is a clique separator if
(i) Q is a clique.
(ii) [V (G) \ Q] is disconnected.
Atom:
A graph is an atom if it does not contain a clique separator.
Clique Separator Decomposition (CSD)
Clique Separator Decomposition (CSD)
• Suppose G has a clique separator C.
Clique Separator Decomposition (CSD)
• Suppose G has a clique separator C.
• Let V (G) = (A, B, C) such that [A, B] = ∅.
Clique Separator Decomposition (CSD)
• Suppose G has a clique separator C.
• Let V (G) = (A, B, C) such that [A, B] = ∅.
• Decompose G into components G0 ∼
= [A ∪ C] and
G00 ∼
= [B ∪ C], separated by C.
Clique Separator Decomposition (CSD)
• Suppose G has a clique separator C.
• Let V (G) = (A, B, C) such that [A, B] = ∅.
• Decompose G into components G0 ∼
= [A ∪ C] and
G00 ∼
= [B ∪ C], separated by C.
• Decompose G0 and G00 further in the same way, and repeat
until no further decomposition is possible.
i.e., we decompose G into a collection of atoms.
Clique Separator Decomposition (CSD)
• Suppose G has a clique separator C.
• Let V (G) = (A, B, C) such that [A, B] = ∅.
• Decompose G into components G0 ∼
= [A ∪ C] and
G00 ∼
= [B ∪ C], separated by C.
• Decompose G0 and G00 further in the same way, and repeat
until no further decomposition is possible.
i.e., we decompose G into a collection of atoms.
• The atoms fit together in hierarchy to form G.
Clique Separator Decomposition (CSD)
Representation of CSD as a Binary Tree (BDT):
External node (Leaf): atom
Internal node: Clique separator
Example
b
a
c
e
d
Example
b
a
c
e
d
Example
b
a
c
e
d
b
a
b
c
e
d
d
Example
b
a
c
e
d
b
a
b
c
{b, d}
e
d
d
{a, b, c, d}
{b, d, e}
Example
b
a
c
e
d
b
b
a
c
{b, d}
e
c
d
{b, c, d}
d
{a, b, c, d}
{b, c, d}
{b, d, e}
R. E. Tarjan
“Decomposition by Clique Separators”
Disc. Math., 55 (1985) 221–232.
• Tarjan [41]: Finding a CSD of G can be done in O(nm)-time.
• Tarjan [41]: CSD can be applied to various optimization
problems; the problem can be solved efficiently on the graph if
it is solvable efficiently on atoms.
Prime Graphs
A subset M ⊆ V (G) is a module of G if every x ∈ V (G) \ M is
either adjacent to all the vertices in in M or to none of the
vertices in M .
G is prime if G has no module M with 1 < |M | < |V (G)|.
V. V. Lozin and M. Milanič
Journal of Discrete Algorithms 6 (2008) 595-604
Let G be a hereditary class of graphs.
If the MWIS problem can be solved in O(np)-time for prime
graphs in G, where p ≥ 1 is a constant, then the MWIS problem
can be solved for graphs in G in time O(np + m).
∴ It is enough to concentrate on Prime Graphs
A. Branstädt and C. T. Hoáng
Theoretical Computer Science 389 (2007) 295-306
Let G be a hereditary class of graphs. Let G ∈ G.
If MWIS problem can be solved in time T = O(f (n)) on prime
atoms of the graph G, then it is solvable in time O(n2 · T ) on G.
∴ It is enough to concentrate on Prime Atoms
Nearly C Property
Let C be a hereditary family of graphs.
A graph G is nearly C if
for every v ∈ V (G), [N (v)] has the property C.
When atoms are nearly C
Suppose that for every vertex v in an atom,
When atoms are nearly C
Suppose that for every vertex v in an atom,
When atoms are nearly C
When atoms are nearly C
Theorem 1:
Let G be a hereditary class of graphs.
Let C be a hereditary property.
Suppose that the atoms of G are nearly C.
If the MWIS problem can be solved in time T = O(f (n)) for
graphs with property C , then the MWIS problem can be solved in
time O(nm + n · T ) for graphs in G.
(P5,
)-free Graphs
MWIS can be solved in O(nm)-time
(P5,
)-free Graphs
MWIS can be solved in O(nm)-time
• Brandstädt and Hoáng [6]: Atoms are nearly chordal.
(P5,
)-free Graphs
MWIS can be solved in O(nm)-time
• Brandstädt and Hoáng [6]: Atoms are nearly chordal.
• Frank [19] : MWIS can be solved in O(m)-time for chordal
graphs.
(P5,
)-free Graphs
MWIS can be solved in O(nm)-time
• Brandstädt and Hoáng [6]: Atoms are nearly chordal.
• Frank [19]: MWIS can be solved in O(m)-time for chordal
graphs.
• So, the results follows by Theorem 1.
MWIS for (P6, C4)-free Graphs
MWIS can be solved in O(nm)-time
(P6, C4)-free Graphs
MWIS can be solved in O(nm)-time
• Brandstädt and Hoáng [6]: Atoms are either nearly chordal or
2-specific graphs.
(P6, C4)-free Graphs
MWIS can be solved in O(nm)-time
• Brandstädt and Hoáng [6]: Atoms are either nearly chordal or
2-specific graphs.
• MWIS is trivial for 2-specific graphs, and can be solved in
O(m)-time for chordal graphs [19].
(P6, C4)-free Graphs
MWIS can be solved in O(nm)-time
• Brandstädt and Hoáng [6]: Atoms are either nearly chordal or
2-specific graphs.
• MWIS is trivial for 2-specific graphs, and can be solved in
O(m)-time for chordal graphs [19].
• So, the result follows by Theorem 1.
(P6,
)-free Graphs5
MWIS in (P6, banner)-free graphs can be solved in O(n3m)-time.
5
T. Karthick, Maximum weight independent sets in (P6 , banner)-free graphs, Submitted for publication.
MWIS for (P6, banner)-free Graphs in O(n3m) time
(1) Prime banner-free graphs are K2,3-free [9].
Prime banner–free graphs are K2, 3 -free
a1
b1
a2
a3
b2
Prime banner–free graphs are K2, 3 -free
a1
b1
a2
a3
b2
Q
Prime banner–free graphs are K2, 3 -free
a1
b1’
a2
a3
z
b2’
Q
Prime banner–free graphs are K2, 3 -free
a1
b1’
a2
a3
z
b2’
Q
Prime banner–free graphs are K2, 3 -free
a1
b1’
a2
a3
z
b2’
Q
Prime banner–free graphs are K2, 3 -free
a1
b1’
a2
a3
z
b2’
Q
Prime banner–free graphs are K2, 3 -free
a1
b1’
a2
a3
z
b2’
Q
Prime banner–free graphs are K2, 3 -free
a1
b1’
a2
a3
z
b2’
Q
Prime banner–free graphs are K2, 3 -free
a1
b1’
a2
a3
z
b2’
Q
MWIS for (P6, banner)-free Graphs in O(n3m) time
(1) Prime banner-free graphs are K2,3-free [9].
(2) MWIS in (P6, P5, banner)-free graphs can be solved in
O(nm)-time.
(P6 ,
Prime,
- free
- free
)-free
(P6 ,
)-free
Assume the Contrary
Prime,
- free
b1
a2
a1
b2
- free
Claim: {b1, b2} is a module
(P6 ,
Prime,
- free
)-free
a1
b1
a2
b2
Claim: {b1, b2} is a module
- free
Assume the contrary,
let pb1E(G) and pb2E(G)
(P6 ,
Prime,
- free
)-free
a1
b1
a2
b2
Claim: {b1, b2} is a module
- free
Then, pa1, pa2 E(G)
(P6 ,
Prime,
- free
)-free
a1
b1
a2
b2
Claim: {b1, b2} is a module
- free
Then, pa1, pa2 E(G)
(P6 ,
Prime,
- free
)-free
a1
b1
a2
b2
Claim: {b1, b2} is a module
- free
Then, pa1, pa2 E(G)
(P6 ,
Prime,
- free
)-free
a1
b1
a2
b2
Claim: {b1, b2} is a module
- free
Then, pa1, pa2 E(G)
(P6 ,
Prime,
- free
- free
)-free
MWIS for prime
(P6, house, banner)-free
graphs can be solved in
O(nm)-time
MWIS for
(P6, C4)-free graphs can be
solved in O(nm)-time
(P6 ,
Prime,
- free
- free
)-free
MWIS for
(P6, house, banner)-free
graphs can be solved in
O(nm)-time
MWIS for
(P6, C4)-free graphs can be
solved in O(nm)-time
MWIS for (P6, banner)-free Graphs in O(n3m) time
(1) Prime banner-free graphs are K2,3-free [9].
(2) MWIS in (P6, P5, banner)-free graphs can be solved in
O(nm)-time.
(3) Brandstädt et al. [9]: By using (1), prime (P6, banner)-free
atoms are nearly C5-free.
Prime (P6, Banner)-free atoms are C5-free
v1
v
v2
v3
v5
A
v4
N(V)
Sketch of the Proof
v1
v
Qv
v2
v3
v5
A
v4
N(V)
Sketch of the Proof
v1
v2
v
v3
Qv
N(A)
v5
A
v4
N(V)
Sketch of the Proof
v1
v2
v
A+
Qv
N(A)
v3
v5
A
v4
N(V)
A1 + = 
v1
v2
v
A1+
Qv
N(A)
v3
v5
A
v4
N(V)
A1 + = 
v1
v2
v
A1+
Qv
N(A)
v3
v5
A
v4
N(V)
A1 + = 
v1
v2
v
A1+
Qv
N(A)
v3
v5
A
v4
N(V)
A2 + = 
v1
v2
v
A2+
Qv
N(A)
v3
v5
A
v4
N(V)
A2 + = 
v1
v2
v
A2+
Qv
N(A)
v3
v5
A
v4
N(V)
A2 + = 
v1
v2
v
A2+
Qv
N(A)
v3
v5
A
v4
N(V)
A2 + = 
v1
v2
v
A2+
Qv
N(A)
v3
v5
A
v4
N(V)
A4 + = 
v1
v2
v
A4+
Qv
N(A)
v3
v5
A
v4
N(V)
A4 + = 
v1
v2
v
A4+
Qv
N(A)
v3
v5
A
v4
N(V)
Sketch of the Proof
v1
A3+
v
A5+
Qv
A+
N(A)
v2
v3
v5
A
v4
N(V)
[A3+, A5+] is complete
v1
v
A3+
A5+
Qv
A+
N(A)
v5
v2
v3
A
v4
N(V)
[A3+, A5+] is complete
v1
v
A3+
A5+
Qv
A+
N(A)
v5
v2
v3
A
v4
N(V)
[A3+, A5+] is complete
v1
v
A3+
A5+
Qv
A+
N(A)
v5
v2
v3
A
v4
N(V)
[A3+, A5+] is complete
v1
v
A3+
A5+
Qv
A+
N(A)
v5
v2
v3
A
v4
N(V)
A5+ is a clique
v1
v
Qv
v5
v2
v3
A5+
N(A)
A
v4
N(V)
A3+ is a clique
v1
v
Qv
v5
v2
v3
A3+
N(A)
A
v4
N(V)
A3+ is a clique
v1
v
Qv
v5
v2
v3
A3+
N(A)
A
v4
N(V)
A3+ is a clique
v1
v
Qv
v5
v2
v3
A
A3+
N(A)
v4
N(V)
A3+ is a clique
v1
v
Qv
v5
v2
v3
A
A3+
N(A)
v4
N(V)
A3+ is a clique
v1
v
Qv
v5
v2
v3
A
A3+
N(A)
v4
N(V)
A3+ is a clique
v1
v2
v
Qv
v5
v3
A
A3+
N(A)
N(V)
v4
A3+ is a clique
v1
v2
v
Qv
v5
v3
A
A3+
N(A)
N(V)
v4
A3+ is a clique
v1
v2
v
Qv
v5
v3
A
A3+
N(A)
N(V)
v4
A3+ is a clique
v1
v2
v
Qv
v5
v3
A
A3+
N(A)
N(V)
v4
A3+ is a clique
v1
v2
v
Qv
v5
v3
A
A3+
N(A)
N(V)
v4
A3+ is a clique
v1
v2
v
Qv
v5
v3
A
A3+
N(A)
N(V)
v4
A3+ is a clique
v1
v2
v
Qv
v5
v3
A
A3+
N(A)
N(V)
v4
A3+ is a clique
v1
v2
v
Qv
v5
v3
v4
A3+
N(A)
N(V)
A3+  A5+ is a clique separator
v1
A3+
v
A5+
Qv
A+
N(A)
v2
v3
v5
A
v4
N(V)
MWIS for (P6, banner)-free Graphs in O(n3m) time
(1) Prime banner-free graphs are K2,3-free [9].
(2) MWIS in (P6, P5, banner)-free graphs can be solved in
O(nm)-time.
(3) Brandstädt et al. [9]: By using (1), prime (P6, banner)-free
atoms are nearly C5-free.
(4) Prime (P6, C5, banner)-free atoms are nearly P5-free.
MWIS for (P6, banner)-free Graphs in O(n3m) time
(1) Prime banner-free graphs are K2,3-free [9].
(2) MWIS in (P6, P5, banner)-free graphs can be solved in
O(nm)-time.
(3) Brandstädt et al. [9]: By using (1), prime (P6, banner)-free
atoms are nearly C5-free.
(4) Prime (P6, C5, banner)-free atoms are nearly P5-free.
(5) By using (2), (4) and Theorem 1, MWIS in (P6, C5, banner)
-free graphs can be solved in O(n2m)-time.
MWIS for (P6, banner)-free Graphs in O(n3m) time
(1) Prime banner-free graphs are K2,3-free [9].
(2) MWIS in (P6, P5, banner)-free graphs can be solved in
O(nm)-time.
(3) Brandstädt et al. [9]: By using (1), prime (P6, banner)-free
atoms are nearly C5-free.
(4) Prime (P6, C5, banner)-free atoms are nearly P5-free.
(5) By using (2), (4) and Theorem 1, MWIS in (P6, C5, banner)
-free graphs can be solved in O(n2m)-time.
(6) By using (3), (5) and Theorem 1, MWIS in (P6, banner)-free
graphs can be solved in O(n3m)-time.
Conclusion
• The complexity of MWIS for P6-free graphs is unknown.
• MWIS remains NP-complete for banner-free graphs [35].
• Showed that MWIS Problem for (P6, banner)-free graphs can
be solved in polynomial time via clique separator
decomposition.
Conclusion contd..
• The complexity of MWIS for P6-free graphs is unknown.
• MWIS remains NP-complete for banner-free graphs [35].
• Showed that MWIS Problem for (P6, banner)-free graphs can
be solved in polynomial time via clique separator
decomposition.
– Time bound has been improved from O(n7m) (given in [9]) to
O(n3m) with the help of atomic structure of given class of
graphs.
Conclusion contd..
• The complexity of MWIS for P6-free graphs is unknown.
• MWIS remains NP-complete for banner-free graphs [35].
• Showed that MWIS Problem for (P6, banner)-free graphs can
be solved in polynomial time via clique separator
decomposition.
– Time bound has been improved from O(n7m) (given in [9]) to
O(n3m) with the help of atomic structure of given class of
graphs.
– This result also generalize known results for (P5, banner)free graphs [6] and for (P6, C4)-free graphs [6].
Conclusion contd..
• Banner-free graphs include some well studied classes of
graphs such as K1,3-free graphs, C4-free graphs, and P4-free
graphs.
• MWIS can be solved in time O(n9m) for (P7, banner)-free
graphs [33].
• MWIS can be solved in time O(n4m) for (P6, co-banner)-free
graphs [34].
• MWIS can be solved in polynomial time for apple-free graphs
[11].
References
[1] V. Alekseev, The effect of local constraints on the complexity of determination of the
graph independence number, Combinatorial-algebraic Methods in Applied Mathematics
(1982) 3-13. (in Russian).
[2] A. Berry, A. Brandstädt, V. Giakoumakis, and F. Maffray, The atomic structure of holeand diamond-free graphs, Manuscript (2011).
[3] A. Brandstädt, (P5, diamond)-free graphs revisited: structure and linear time
optimization, Discrete Applied Mathematics, 138 (2004) 13-27.
[4] A. Brandstädt and D. Kratsch, On the structure of (P5, Gem)-free graphs, Discrete
Applied Mathematics, 145 (2005) No.2, 155-162.
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