A subgraph of an edge-coloured graph is rainbow if all of its edges have different colours. For graphs G and H the anti-Ramsey number ar(G,H) is the maximum number of colours in an edge-colouring of G with no rainbow copy of H. The notion was introduced by Erdős, Simonovits and V. Sós and studied in case G = Kn. Afterwards exact values or bounds for anti-Ramsey numbers ar(Kn, H) were establishe...

Blbzsik, Z., M. Hujter, A. Pluhir and Z. Tuza, Graphs with no induced C4 and 2K,, Discrete Mathematics 115 (1993) 51-55. We characterize the structure of graphs containing neither the 4-cycle nor its complement as an induced subgraph. This self-complementary class B of graphs includes split graphs, which are graphs whose vertex set is the union of a clique and an independent set. In the members...

Consider a graph G which belongs to class C. We are interested in connecting node w∉V(G) by single edge uw where u∈V(G); we call such an tail. As the resulting from after addition of tail, denoted G+uw, need not belong C, want compute number non-edges minimum C-completion i.e., (excluding tail uw) be added G+uw so that In this paper, study problem for classes split, quasi-threshold, threshold a...

Parity graphs form a superclass of bipartite and distance-hereditary graphs. Since their introduction, all the algorithms proposed as solutions to the recognition problem and other combinatorial problems exploit the structural property of these graphs described by Burlet and Uhry in [8]. This paper introduces a different structural property of parity graphs: split decomposition returns exactly,...

In this paper we introduce a generalization of the c-planarity testing problem for clustered graphs. Namely, given a clustered graph, the goal of the SPLIT-C-PLANARITY problem is to split as few clusters as possible in order to make the graph c-planar. Determining whether zero splits are enough coincides with testing c-planarity. We show that SPLIT-C-PLANARITY is NP-complete for c-connected clu...

The Cops and Robbers game is played on undirected graphs where a group of cops tries to catch a robber. The game was defined independently by Winkler-Nowakowski and Quilliot in the 1980s and since that time has been studied intensively. Despite of that, its computation complexity is still an open question. In this paper we prove that computing the minimum number of cops that can catch a robber ...

An (r, n)-split coloring of a complete graph is an edge coloring with r colors under which the vertex set is partitionable into r parts so that for each i, part i does not contain K, in color i. This generalizes the notion of split graphs which correspond to (2, 2)-split colorings. The smallest N for which the complete graph Ks has a coloring which is not (r,n)-split is denoted by f,.(n). Balan...

The concept of connectedness plays an important role in many networks. Digraphs are considered as an excellent modeling tool and are used to model many types of relations amongst any physical situations. In this paper the concept of strong non-split domination in directed graph D has been introduced by considering the dominating set S is a strong non-split dominating set if the complement of S ...

A vertex ranking of a graph G is an assignment of positive integers (colors) to the vertices of G such that each path connecting two vertices of the same color contains a vertex of a higher color. Our main goal is to find a vertex ranking using as few colors as possible. Considering on-line algorithms for vertex ranking of split graphs, we prove that the worst case ratio of the number of colors...

SUMMARY The design and implementation of an interference graph is critical to the performance of a graph-coloring register allocator. The cost of constructing and manipulating the interference graph dominates the overall cost of allocation. The literature on graph-coloring register allocation suggests the use of a bit matrix coupled with lists of edges to represent the graph. 1-3 Recently, Geor...