New properties of perfectly orderable graphs and strongly perfect graphs (original) (raw)
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A note on perfectly orderable graphs
Discrete Applied Mathematics, 1996
We introduce a new class of perfectly orderable graphs that contains complements of chordal bipartite graphs, unions of two threshold graphs. graphs with Dilworth number at most three. and complements of triangulated graphs.
Some classes of perfectly orderable graphs
Journal of Graph Theory, 1989
In 1981, Chvátal defined the class of perfectly orderable graphs. This class of perfect graphs contains the comparability graphs and the triangulated graphs. In this paper, we introduce four classes of perfectly orderable graphs, including natural generalizations of the comparability and triangulated graphs. We provide recognition algorithms for these four classes. We also discuss how to solve the clique, clique cover, coloring, and stable set problems for these classes.
A charming class of perfectly orderable graphs
Discrete Mathematics, 1992
We investigate the following conjecture of VaSek Chvatal: any weakly triangulated graph containing no induced path on five vertices is perfectly orderable. In the process we define a new polynomially recognizable class of perfectly orderable graphs called charming. We show that every weakly triangulated graph not containing as an induced subgraph a path on five vertices or the complement of a path on six vertices is charming.
On P4-transversals of perfect graphs
Discrete Mathematics, 2000
A subset T of vertices of a graph G is called a P4-transversal if T meets every P4 of G; if in addition T induces a P4-free subgraph then T is called a two-sided P4-transversal of G. We conjecture that graphs containing a two-sided P4-transversal are perfect provided they contain no odd chordless cycle with at least ÿve vertices or the complement of such a cycle. We show that, as a consequence of previously known results, this conjecture is true for graphs containing a stable P4-transversal. We show, using a reduction from 3-SAT, that it is NP-complete to decide if a perfect graph has a stable P4-transversal. Next, we prove that a graph is perfectly orderable, in the sense of Chvà atal, if it has a stable set meeting all P4's in an end-point or all P4's in a mid-point. We show that there is a polynomial time algorithm to recognize such a graph using a reduction to 2-SAT. Finally, we prove that our conjecture is true if the P4-transversal induces a threshold graph and meets every P4 in an end-point, or meets every P4 in a mid-point.
Some properties of minimal imperfect graphs
Discrete Mathematics, 1996
The Even Pair Lemma, proved by Meyniel, states that no minimal imperfect graph contains a pair of vertices such that all chordless paths joining them have even lengths. This Lemma has proved to be very useful in the theory of perfect graphs. The Odd Pair Conjecture, with 'even' replaced by 'odd', is the natural analogue of the Even Pair Lemma. We prove a partial result for this conjecture, namely: no minimal imperfect graph G contains a three-pair, i.e. two nonadjacent vertices Ul, u2 such that all chordless paths of G joining ul to u2 contain precisely three edges. As a by-product, we obtain short proofs of two previously known theorems: the first one is a well-known theorem of Meyniel (a graph is perfect if each of its odd cycles with at least five vertices contains at least two chords), the second one is a theorem of Olariu (a graph is perfect if it contains no odd antihole, no Ps and no extended claw as induced subgraphst.
Discrete Mathematics, 1989
Perfectly orderable graphs were introduced by Chvfital in 1984. Since then, several classes of perfectly orderable graphs have been identified. In this paper, we establish three new results on perfectly orderable graphs. First, we prove that every graph with Dilworth number at most three has a simplicial vertex, in the graph or in its complement. Second, we provide a characterization of graphs G with this property: each maximal vertex of G is simplicial in the complement of G. Finally, we introduce the notion of a locally perfect order and show that every arborescence-comparability graph admits a locally perfect order.
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.
P A SUPER STRONGLY PERFECTNESS OF SOME GRAPHS
A Graph G is Super Strongly Perfect Graph if every induced sub graph H of G possesses a minimal dominating set that meets all the maximal cliques of H. The structure of Super Strongly Perfect Graphs have been characterized by some classes of graphs like Cycle graphs, Circulant graphs, Complete graphs, Complete Bipartite graphs etc., In this paper, we have analysed some other graph classes like, Bicyclic graphs, Dumb bell graphs and Star graphs to characterize the structure of Super Strongly Perfect Graphs in a different way. By this we found the cardinality of minimal dominating set and maximal cliques of the above graphs.
Some variations of perfect graphs
Discussiones Mathematicae Graph Theory, 2016
We consider (ψ k −γ k−1)-perfect graphs, i.e., graphs G for which ψ k (H) = γ k−1 (H) for any induced subgraph H of G, where ψ k and γ k−1 are the k-path vertex cover number and the distance (k − 1)-domination number, respectively. We study (ψ k −γ k−1)-perfect paths, cycles and complete graphs for k ≥ 2. Moreover, we provide a complete characterisation of (ψ 2 − γ 1)perfect graphs describing the set of its forbidden induced subgraphs and providing the explicit characterisation of the structure of graphs belonging to this family.