On the complexity of recognizing a class of perfectly orderable graphs (original) (raw)

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.

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.

The n -ordered graphs: A new graph class

Journal of Graph Theory, 2009

For a positive integer n, we introduce the new graph class of n-ordered graphs, which generalize partial n-trees. Several characterizations are given for the finite n-ordered graphs, including one via a combinatorial game.

ON SEMI-TRANSITIVE ORIENTATIONS AND GRAPHS REPRESENTABLE BY WORDS

2010

A graph G = (V, E) is representable if there exists a word W over the alphabet V such that letters x and y alternate in W if and only if (x, y) ∈ E for each x = y. If W is k-uniform (each letter of W occurs exactly k times in it) then G is called k-representable. The minimum k for which a representable graph G is k-representable is called its representation number.

Making arbitrary graphs transitively orientable: Minimal comparability completions

2006

A transitive orientation of an undirected graph is an assignment of directions to its edges so that these directed edges represent a transitive relation between the vertices of the graph. Not every graph has a transitive orientation, but every graph can be turned into a graph that has a transitive orientation, by adding edges. We study the problem of adding an inclusion minimal set of edges to an arbitrary graph so that the resulting graph is transitively orientable. We show that this problem can be solved in polynomial time, and we give a surprisingly simple algorithm for it.

A class of ?-perfect graphs

Discrete Mathematics, 2000

Consider the following total order: order the vertices by repeatedly removing a vertex of minimum degree in the subgraph of vertices not yet chosen and placing it after all the remaining vertices but before all the vertices already removed. For which graphs the greedy algorithm on this order gives an optimum vertex-coloring? Markossian, Gasparian and Reed introduced the class of ÿ-perfect graphs. These graphs admit such a greedy vertex-coloring algorithm. The recognition of ÿ-perfect graphs is open. We deÿne a subclass of ÿ-perfect graphs, that can be recognized in polynomial time, by considering the class of graphs with no even hole, no short-chorded cycle on six vertices, and no diamond. In particular, we make use of the following properties: no minimal ÿ-imperfect graph contains a simplicial vertex, a minimal ÿ-imperfect graph which is not an even hole contains no vertex of degree 2.

A Polynomial Algorithm for Recognizing Perfect Graphs

2003

We present a polynomial algorithm for recognizing whether a graph is perfect, thus settling a long standing open question. The algorithm uses a decomposition theorem of Conforti, Cornuéjols and Vušković. Another polynomial algorithm for recognizing perfect graphs, which does not use decomposition, was obtained simultaneously by Chudnovsky and Seymour. Both algorithms need a first phase developed jointly by Chudnovsky, Cornuéjols, Liu, Seymour and Vušković.

Minimal comparability completions of arbitrary graphs

Discrete Applied Mathematics, 2008

A transitive orientation of an undirected graph is an assignment of directions to its edges so that these directed edges represent a transitive relation between the vertices of the graph. Not every graph has a transitive orientation, but every graph can be turned into a graph that has a transitive orientation, by adding edges. We study the problem of adding an inclusion minimal set of edges to an arbitrary graph so that the resulting graph is transitively orientable. We show that this problem can be solved in polynomial time, and we give a surprisingly simple algorithm for it. We use a vertex incremental approach in this algorithm, and we also give a more general result that describes graph classes Π for which Π completion of arbitrary graphs can be achieved through such a vertex incremental approach.

New properties of perfectly orderable graphs and strongly perfect graphs

Discrete Mathematics, 1991

Hoang, C.T., F. Maffray and M. Preissmann, New properties of perfectly orderable graphs and strongly perfect graphs, Discrete Mathematics 98 (1991) 161-174. We establish a property of minimal nonperfectly orderable graphs, and use this property to generate a class of perfectly orderable graphs which strictly contains all brittle graphs. This class is characterized by the existence, in each induced subgraph, of a vertex which is either the endpoint of no Ps, or the midpoint of no P4, or the mid-point of exactly one P4 and the endpoint of exactly one P4. As a consequence, we show that the number of P4's in a minimal nonperfectly orderable graph is at least in, where n is the number of vertices of the graph. Similar results are obtained for strongly perfect graphs.