In this paper, we consider the mutual exclusion scheduling problem for comparability graphs. Given an undirected graph G and a xed constant m, the problem is to nd a minimum coloring of G such that each color is used at most m times. The complexity of this problem for comparability graphs was mentioned as an open problem by M ohring (1985) and for permutation graphs (a subclass of comparabilit...

A k-subcoloring of a graph is a partition of the vertex set into at most k cluster graphs, that is, graphs with no induced P3. 2-subcoloring is known to be NP-complete for comparability graphs and three subclasses of planar graphs, namely triangle-free planar graphs with maximum degree 4, planar perfect graphs with maximum degree 4, and planar graphs with girth 5. We show that 2-subcoloring is ...

In this paper, we present a linear-time algorithm for substitution decomposition on chordal graphs. Based on this result, we develop a linear-time algorithm for transitive orientation on chordal comparability graphs, which reduces the complexity of chordal comparability recognition from O(n) to O(n+m). We also devise a simple linear-time algorithm for interval graph recognition where no complic...

Hajnal, A. and N. Sauer, Cut-sets in infinite graphs and partial orders. Discrete Mathematics 117 (1993) 113-125. The set S c V(U) is a cut-set of the vertex v of a graph 9 if v is not adjacent to any vertex in S and, for every maximal clique C of Q, ({v} u S) n C # 0. S is a cut-set of the element v of a partial order 9 if S is a cut-set of v in the comparability graph of 8. Given upper bounds...

A nonnegative matrix T = (tij)~,j= 1 is a generalized transitive tournament matrix (GTT matrix) if tii = 0, t o = 1 tj~ for i # j , and 1 ~< t o + tjk + t~ ~< 2 for i, j , k pairwise distinct. The problem we are interested in is the characterization of the set of vertices of the polytope {GTT}n of all GTT matrices of order n. In 1992, Brualdi and Hwang introduced the *-graph associated to each ...

We introduce a notion of simultaneity for any class of graphs with an intersection representation (interval graphs, chordal graphs, etc.) and for comparability graphs, which are represented by transitive orientations. Let G1 and G2 be graphs from such a class C, sharing some vertices I and the corresponding induced edges. Then G1 and G2 are said to be simultaneous C graphs if there exist repres...

Let Kbe a set, finite or infinite. A simple directed graph P = {V, U) is said to be "transitively oriented" if the existence of a directed path from a vertex x to a vertex y implies the existence of an arc (x, y)e U. For undefined graph-theoretical terms see for instance [4]. A transitively oriented graph on the vertex set V is nothing but the graph of some partial ordering of the set V; loops ...

This paper deals with the graph isomorphism (GI) problem for two graph classes: chordal bipartite graphs and strongly chordal graphs. It is known that GI problem is GI complete even for some special graph classes including regular graphs, bipartite graphs, chordal graphs, comparability graphs, split graphs, and k-trees with unbounded k. On the other side, the relative complexity of the GI probl...

The scheduling and mapping of task graph to processors is considered to be the most crucial NP-complete in parallel and distributed computing systems. In this paper, the theoretical graph application using matching is presented to assign a number of tasks onto two processors. This paper addresses a directed-weighted cyclic graph. The effort is to reduce the graph onto directed acyclic graph. A ...

The composition graph of a family of n+1 disjoint graphs {Hi : 0 ≤ i ≤ n} is the graph H obtained by substituting the n vertices of H0 respectively by the graphs H1, H2, ..., Hn. If H has some hereditary property P , then necessarily all its factors enjoy the same property. For some sort of graphs it is sufficient that all factors {Hi : 0 ≤ i ≤ n} have a certain common P to endow H with this P ...