Showing papers on "Split graph published in 1977"
TL;DR: The domatic number of a graph is defined and studied and it is seen that the theory of domination resembles the well known theory of colorings of graphs.
Abstract: This paper presents a quick review of results and applications concerning dominating sets in graphs. The domatic number of a graph is defined and studied. It is seen that the theory of domination resembles the well known theory of colorings of graphs.
542 citations
TL;DR: It is proved that Berge's Strong Perfect Graph Conjecture is valid for 3-chromatic graphs and not just for perfect graphs based on Fulkerson's antiblocking polyhedra approach to perfect graphs.
72 citations
TL;DR: The purpose of this paper is to describe polynomial time algorithms for the recognition of the clique separable graphs and for finding them a minimum coloring and a maximum clique.
52 citations
TL;DR: It is shown that there exists a linear algorithm for constructing a Hamiltonian circuit in a connected (3, 1) graph and an n 4 -algorithm for finding a minimum coloring in a (3-1) graph.
1 citations
TL;DR: The authors' counting method is based on a characterization of the graphs which can be formed when a line is deleted from a 3-connected graph, and can be used to derive the recursive formulae of Read for the numbers of labelled cubic graphs and labelled connected cubic graphs.
Abstract: Labelled connected graphs were first counted by Riddell [3], as were unlabelled connected graphs and labelled 2-connected graphs. Robinson [4] developed the only known method of counting unlabelled 2-connected graphs. We have found a recursive method of counting labelled 3connected graphs. The corresponding problem for unlabelled graphs remains unsolved. Our counting method is based on a characterization of the graphs which can be formed when a line is deleted from a 3-connected graph. Consequently, it is necessary to include the number of lines as a parameter of the enumeration. The characterization result also allows US to count labelled 3-connected cubic graphs. Read [l] and [2] has already counted labelled cubic graphs and labelled connected cubic graphs. Since no cubic graph is 4-connected, it only remained to enumerate labelled 2-connected cubic graphs. We have accomplished this by taking into consideration graphs which are cubic except for one or two points of degree 2. The method can also be used to derive the recursive formulae of Read [2] for the numbers of labelled cubic graphs and labelled connected cubic graphs. In addition, we have used related techniques to show that almost all labelled cubic graphs are 3-connected, the analog for cubic graphs of the fact'that almost all labelled graphs are 3-connected.