Polar cographs Tınaz Ekim 1 N.V.R. Mahadev 2 Dominique de Werra 1 1 IMA-ROSE Ecole Polytechnique Fédérale de Lausanne, Switzerland tinaz.ekim@epfl.ch, dewerra.ima@epfl.ch 2 Fitchburg State College, USA nmahadev@fsc.edu July 2006
Outline Definitions 1 Definitions Polar graphs Cographs 2
(s, k)-polar graph Definitions Polar graphs Cographs A graph G = (V, E) is (s, k)-polar if R, B such that 1 R B = and R B = V 2 R induces a complete s-partite graph 3 B induces disjoint union of k cliques
(s, k)-polar graph Definitions Polar graphs Cographs A graph G = (V, E) is (s, k)-polar if R, B such that 1 R B = and R B = V 2 R induces a complete s-partite graph 3 B induces disjoint union of k cliques 4 χ(g[r]) = s and R is P 3 -free 5 θ(g[b]) = k and B is P 3 -free
Polar graphs Cographs Remark Not every graph is polar.
Polar graphs Cographs Remark Not every graph is polar. Figure: A minimal non-polar graph.
Polar graphs Cographs Remark Not every graph is polar. Figure: A possible polar partition of G \ v.
State of the art Definitions Polar graphs Cographs Recognition of polar graphs is NP-hard (Chernyak, Chernyak, 1986) Polynomial cases where the sizes of cliques and/or stable sets are/is bounded (Mel nokov, Kozhich 1985), (Tyshkevich, Chernyak 1985) If s and k are fixed, then recognizing polar perfect graphs is in P (Feder et al. 2004), (Alekseev et al. 2004)
Cographs Definitions Polar graphs Cographs P 4 -free Recursively constructed with two operations: join disjoint union Cotree representation: 1-vertex, 0-vertex 1 0 0 1 1 e f e f a b c d Either G or G is disconnected a b c d
Polar cographs 1 Definitions Theorem For a cograph G, the following statements are equivalent: a) G is polar. b) Neither G nor G contains any one of the graphs H 1,..., H 4 as induced subgraphs. H 1 H 2 H 3 H 4
Polar cographs (Sketch of proof) 2 a) b) Suppose H 1,..., H 4 are polar. Complete join of stable sets is connected one component is a disjoint union of cliques. Contradiction! H 1 H 2 H 3 H 4
Polar cographs (Sketch of proof) 3 b) a) Suppose G (minimal) non-polar and disconnected. G[A] and G[B] are polar P 3 G[A] and P 3 G[B] G[A] has at least 2 stable sets in every polar partition G A B
Polar cographs (Sketch of proof) 3 b) a) Suppose G (minimal) non-polar and disconnected. G[A] and G[B] are polar P 3 G[A] and P 3 G[B] G[A] has at least 2 stable sets in every polar partition G A B
Polar cographs (Sketch of proof) 3 b) a) Suppose G (minimal) non-polar and disconnected. G[A] and G[B] are polar P 3 G[A] and P 3 G[B] G[A] has at least 2 stable sets in every polar partition G A B
Polar cographs (Sketch of proof) 4 b) a) A A\A A B Let A A induce the connected component containing the join of stable sets A \ A is a disjoint union of cliques
Polar cographs (Sketch of proof) 4 b) a) A A\A A B Let A A induce the connected component containing the join of stable sets A \ A is a disjoint union of cliques A connected + G cograph G[A ] = G[C] G[D] Case 1 : P 3 in A such that P 3 D Contradiction! Case 2 : P 3 of A intersects both C and D
Monopolar cographs Definitions Theorem For a cograph G, the following are equivalent. a) G is monopolar. b) Neither G nor G contains any one of the graphs G 1,..., G 9 as an induced subgraph. c) G or G is a disjoint union of threshold graphs and complete (1, )-polar graphs. G 1 G 2 G 3 G 4 G 5 G 6 G 7 G 8 G 9
Maximum induced polar subgraph Theorem For any cograph G given by its cotree, a maximum induced polar subgraph can be computed in time O(n 2 ). Dynamic programming on the cotree to compute at each vertex: Max Stable Set, Max Clique, Max Threshold graph O(n) Max Union of Cliques, Max join of stable sets O(n) Max Monopolar Stable, Max Monopolar Clique O(n 2 ) Max Polar O(n 2 ) recognition of polar cographs in time O(n 2 ).
1 Lemma A connected (1, )-polar cograph which is not a clique or a threshold graph, is a complete (1, )-polar graph. This can be recognized in linear time.. clique threshold graph complete (1, )-polar
2 Theorem For any cograph G, it can be recognized whether G is a polar graph in time O(n log n). At a 0-vertex x, G(x) is polar 0 0. OR clique threshold graph complete (1, ) polar polar graph with at least 2 stable sets clique disjoint union of cliques
Future research Definitions Polar chordal graphs (Ekim, Hell, Stacho, de Werra, 2006) P 4 -reducible graphs Permutation graphs
Future research Definitions Polar chordal graphs (Ekim, Hell, Stacho, de Werra, 2006) P 4 -reducible graphs Permutation graphs Thank you for your attention