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http://dx.doi.org/10.4134/JKMS.2005.42.6.1251

GRAPHS WITH ONE HOLE AND COMPETITION NUMBER ONE  

KIM SUH-RYUNG (Department of Mathematics Education Seoul National University)
Publication Information
Journal of the Korean Mathematical Society / v.42, no.6, 2005 , pp. 1251-1264 More about this Journal
Abstract
Let D be an acyclic digraph. The competition graph of D has the same set of vertices as D and an edge between vertices u and v if and only if there is a vertex x in D such that (u, x) and (v, x) are arcs of D. The competition number of a graph G, denoted by k(G), is the smallest number k such that G together with k isolated vertices is the competition graph of an acyclic digraph. It is known to be difficult to compute the competition number of a graph in general. Even characterizing the graphs with competition number one looks hard. In this paper, we continue the work done by Cho and Kim[3] to characterize the graphs with one hole and competition number one. We give a sufficient condition for a graph with one hole to have competition number one. This generates a huge class of graphs with one hole and competition number one. Then we completely characterize the graphs with one hole and competition number one that do not have a vertex adjacent to all the vertices of the hole. Also we show that deleting pendant vertices from a connected graph does not change the competition number of the original graph as long as the resulting graph is not trivial, and this allows us to construct infinitely many graph having the same competition number. Finally we pose an interesting open problem.
Keywords
competition number; chordal graph; chordless cycle; hole;
Citations & Related Records

Times Cited By Web Of Science : 9  (Related Records In Web of Science)
Times Cited By SCOPUS : 8
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