On (P5, diamond)-free graphs (original) (raw)
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On graphs whose maximal cliques and stable sets intersect
We say that a graph G has the CIS-property and call it a CIS-graph if every maximal clique and every maximal stable set of G intersect. By definition, G is a CIS-graph if and only if the complementary graph G is a CISgraph. Let us substitute a vertex v of a graph G ′ by a graph G ′′ and denote the obtained graph by G. It is also easy to see that G is a CIS-graph if and only if both G ′ and G ′′ are CIS-graphs. In other words, CIS-graphs respect complementation and substitution. Yet, this class is not hereditary, that is, an induced subgraph of a CIS-graph may have no CIS-property. Perhaps, for this reason, the problems of efficient characterization and recognition of CIS-graphs are difficult and remain open. In this paper we only give some necessary and some sufficient conditions for the CIS-property to hold. There are obvious sufficient conditions. It is known that P 4-free graphs have the CIS-property and it is easy to see that G is a CIS-graph whenever each maximal clique of G has a simplicial vertex. However, these conditions are not necessary. There are also obvious necessary conditions. Given an integer k ≥ 2, a comb (or k-comb) S k is a graph with 2k vertices k of which, v 1 ,. .. , v k , form a clique C, while others, v ′ 1 ,. .. , v ′ k , form a stable set S, and (v i , v ′ i) is an edge for all i = 1,. .. , k, and there are no other edges. The complementary graph S k is called an anti-comb (or k-anti-comb). Clearly, S and C switch in the complementary graphs. Obviously, the combs and anti-combs are not CIS-graphs, since C ∩ S = ∅. Hence, if a CIS-graph G contains an induced comb or anti-comb then it must be settled, that is, G must contain a vertex v connected to all vertices of C and to no vertex of S. However, these conditions are only necessary. The following sufficient conditions are more difficult to prove: G is a CIS-graph whenever G contains no induced 3-combs and 3-anti-combs, and every induced 2comb is settled in G. It is an open question whether G is a CIS-graph if G contains no induced 4-combs and 4-anti-combs, and all induced 3-combs, 3-anti-combs, and 2-combs are settled in G. We generalize the concept of CIS-graphs as follows. For an integer d ≥ 2 we define a d-graph G = (V ; E 1 ,. .. , E d) as a complete graph whose edges are colored by d colors (that is, partitioned into d sets). We say that G is a CIS-d-graph (has the CIS-d-property) if d i=1 C i = ∅ whenever for each i = 1,. .. , d the set C i is a maximal color i-free subset of V , that is, (v, v ′) ∈ E i for any v, v ′ ∈ C i. Clearly, in case d = 2 we return to the concept of CIS-graphs. (More accurately, CIS-2-graph is a pair of two complementary CIS-graphs.) We conjecture that each CIS-d-graph is a Gallai graph, that is, it contains no triangle colored by 3 distinct colors. We obtain results supporting this conjecture and also show that if it holds then characterization and recognition of CIS-d-graphs are easily reduced to characterization and recognition of CIS-graphs. We also prove the following statement. Let G = (V ; E 1 ,. .. , E d) be a Gallai d-graph such that at least d − 1 of its d chromatic components are CIS-graphs, then G has the CIS-d-property. In particular, the remaining chromatic component of G is a CIS-graph too. Moreover, all 2 d unions of d chromatic components of G are CISgraphs.
Clique-perfectness and balancedness of some graph classes
International Journal of Computer Mathematics, 2014
A graph is clique-perfect if the maximum size of a clique-independent set (a set of pairwise disjoint maximal cliques) and the minimum size of a clique-transversal set (a set of vertices meeting every maximal clique) coincide for each induced subgraph. A graph is balanced if its clique-matrix contains no square submatrix of odd size with exactly two ones per row and column. In this work, we give lineartime recognition algorithms and minimal forbidden induced subgraph characterizations of cliqueperfectness and balancedness of P 4 -tidy graphs and a linear-time algorithm for computing a maximum clique-independent set and a minimum clique-transversal set for any P 4 -tidy graph. We also give a minimal forbidden induced subgraph characterization and a linear-time recognition algorithm for balancedness of paw-free graphs. Finally, we show that clique-perfectness of diamond-free graphs can be decided in polynomial time by showing that a diamond-free graph is clique-perfect if and only if it is balanced.
On color-critical ($P_{5},\overline{P}_5$)-free graphs
Cornell University - arXiv, 2014
A graph is k-critical if it is k-chromatic but each of its proper induced subgraphs is (k − 1)-colorable. It is known that the number of 4-critical P 5-free graphs is finite, but there is an infinite number of k-critical P 5-free graphs for each k ≥ 5. We show that the number of k-critical (P 5 , P 5)-free graphs is finite for every fixed k. Our result implies the existence of a certifying algorithm for k-coloring (P 5 , P 5)-free graphs.
On Tucker's proof of the strong perfect graph conjecture for -free graphs
Discrete Mathematics, 2001
In this note, the authors generalize the ideas presented by Tucker in his proof of the Strong Perfect Graph Conjecture for (K4 − e)-free graphs in order to ÿnd a vertex v in G whose special neighborhood allows to extend a !(G)-vertex coloring of G − v to a !(G)-vertex coloring of G.
On α-redundant vertices in P5-free graphs
Information Processing Letters, 2002
We prove that MAXIMUM STABLE SET can be solved in polynomial time on two new subclasses of P 5 -free graphs, extending some known polynomially solvable cases.
On Algorithms for (P5,Gem)-Free Graphs
2003
A graph is (P 5 ,gem)-free, when it does not contain P 5 (an induced path with five vertices) or a gem (a graph formed by making an universal vertex adjacent to each of the four vertices of the induced path P 4 ) as an induced subgraph.
A new characterization of trivially perfect graphs
Electronic Journal of Graph Theory and Applications, 2015
A graph G is trivially perfect if for every induced subgraph the cardinality of the largest set of pairwise nonadjacent vertices (the stability number) α(G) equals the number of (maximal) cliques m(G). We characterize the trivially perfect graphs in terms of vertex-coloring and we extend some definitions to infinite graphs.
The Star and Biclique Coloring and Choosability Problems
Journal of Graph Algorithms and Applications, 2014
A biclique of a graph G is an induced complete bipartite graph. A star of G is a biclique contained in the closed neighborhood of a vertex. A star (biclique) k-coloring of G is a k-coloring of G that contains no monochromatic maximal stars (bicliques). Similarly, for a list assignment L of G, a star (biclique) L-coloring is an L-coloring of G in which no maximal star (biclique) is monochromatic. If G admits a star (biclique) Lcoloring for every k-list assignment L, then G is said to be star (biclique) k-choosable. In this article we study the computational complexity of the star and biclique coloring and choosability problems. Specifically, we prove that the star (biclique) k-coloring and k-choosability problems are Σ p 2-complete and Π p 3-complete for k > 2, respectively, even when the input graph contains no induced C4 or K k+2. Then, we study all these problems in some related classes of graphs, including H-free graphs for every H on three vertices, graphs with restricted diamonds, split graphs, and threshold graphs.
P5-free augmenting graphs and the maximum stable set problem
Discrete Applied Mathematics, 2003
The complexity status of the maximum stable set problem in the class of P5-free graphs is unknown. In this paper, we ÿrst propose a characterization of all connected P5-free augmenting graphs. We then use this characterization to detect families of subclasses of P5-free graphs where the maximum stable set problem has a polynomial time solution. These families extend several previously studied classes. ?