Showing papers on "Path graph published in 1972"
01 Mar 1972
TL;DR: In this paper, an algorithm for finding a maximum clique in an arbitrary graph is described, which has a worst-case time bound of k(1.286) sup N for some constant k, where n is the number of vertices in the graph.
Abstract: : An algorithm for finding a maximum clique in an arbitrary graph is described. The algorithm has a worst-case time bound of k(1.286) sup N for some constant k, where n is the number of vertices in the graph. Within a fixed time, the algorithm can analyze a graph with 2 3/4 as many vertices as the largest graph which the obvious algorithm (examining all subsets of vertices) can analyze. (Author)
29 citations
TL;DR: A graph of genus 2 which is irreducible with respect to this property must have at least eight vertices as discussed by the authors, and it is shown that there are exactly three such 2-irreducibly connected graphs having 8 vertices, and that these three graphs, together with the Kuratowski theorem, leeds to a determination of the genus of each graph having fewer than nine vertices.
Abstract: A graph of genus 2 which is irreducible with respect to this property must have at least eight vertices. It is shown here that there are exactly three such 2-irreducible graphs having eight vertices. The description of these three graphs, together with the Kuratowski theorem, leeds to a determination of the genus of each graph having fewer than nine vertices.
13 citations
01 Jan 1972
TL;DR: In this paper, the authors describe the complete tripartite graph K a,b,c, and denote its path number by k(a, b, c) by assuming that abc ≠ 0 and with this restriction two obvious lemmas may be obtained.
Abstract: Publisher Summary This chapter describe the complete tripartite graph K a,b,c , and denote its path number by k(a, b, c). By assuming that abc ≠ 0 and with this restriction two obvious lemmas may be obtained. This chapter also focuses on the vector of integers in nonincreasing order. k (υ) is defined as the path number to complete η -partite graph on (υ 1 , υ2, …, υ n ) vertices, and was found that many of the results were generalized.
10 citations
TL;DR: In this paper, it was shown that a strong complete graph on n(>3) vertices has at most n-2 cut vertices, without using the existence of a Hamiltonian circuit.
Abstract: In this paper we determine the ranges of the number of cut vertices and the number of cut arcs in a strong graph on n vertices with m arcs Also it is proved that a strong complete graph on n(>3) vertices has at most n-2 cut vertices, without using the existence of a Hamiltonian circuit, thus solving a problem posed by KORVIN
3 citations
Book•
01 Jan 1972
1 citations
1 citations
01 Jan 1972
TL;DR: In this article, the existence of double points in certain tournaments is shown to imply that a vertex x E a, is a double point provided there exist two antidirected Hamiltonian paths in rt-,, both starting with x.
Abstract: The notion of an antidirected Hamiltonian path was introduced by Griinbaum in [1] and defined as follows: A simple path in a directed graph is antidirected provided every two adjacent edges of the path have opposing orientations. A path is Hamiltonian provided it is simple and contains all the vertices of the graph. An ADH path is an antidirected Hamiltonian path in a directed graph. An n-tournament T, is an oriented complete grapb with n vertices. Our proof of Griinbaum’s theorem is based on the existence of double points in certain tournaments. A vertex x E a, is a double point provided there exist two ADH paths in rt-, , both starting with x, as follows: