Properly orderable graphs

Properly orderable graphs

In a graph G = (V, E) provided with a linear order ”<” on V , a chordless path with vertices a, b, c, d and edges ab, bc, cd is called an obstruction if both a < b and d < c hold. Chvátal [2] defined the class of perfectly orderable graphs (i.e., graphs possessing an acyclic orientation of the edges such that no obstruction is induced) and proved that they are perfect. We introduce here the cla...

Homogeneously Orderable Graphs

In this paper we introduce homogeneously orderable graphs which are a common generalization of distance-hereditary graphs, dually chordal graphs and homogeneous graphs. We present a characterization of the new class in terms of a tree structure of the closed neighborhoods of homogeneous sets in 2-graphs which is closely related to the defining elimination ordering. Moreover, we characterize the...

Uniquely Partially Orderable Graphs

A charming class of perfectly orderable graphs

Hoang, C.T., F. Maffray, S. Olariu and M. Preissmann, A charming class of perfectly orderable graphs, Discrete Mathematics 102 (1992) 67-74. 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 ...

Perfectly Orderable Graphs and Unique Colorability

Given a linear order < on the vertices of a graph, an obstruction is an induced P4 abcd such that a < b and d < c. A linear order without any obstruction is called perfect. A graph is perfectly orderable if its vertex set has some perfect order. In the graph G, for two vertices x and y, x clique-dominates y if every maximum size clique containing y, contains x too. We prove the following result...