Abstract
Graph Theory In 1982, Opsut showed that the competition number of a line graph is at most two and gave a necessary and sufficient condition for the competition number of a line graph being one. In this paper, we generalize this result to the competition numbers of generalized line graphs, that is, we show that the competition number of a generalized line graph is at most two, and give necessary conditions and sufficient conditions for the competition number of a generalized line graph being one.
Highlights
The notion of a competition graph was introduced by Cohen [1] as a means of determining the smallest dimension of ecological phase space
We investigate the competition number of a generalized line graph which was introduced by Hoffman [2] in 1970
In Subsection 2.3, we investigate generalized line graphs whose competition numbers are one, and give some sufficient conditions and necessary conditions
Summary
The notion of a competition graph was introduced by Cohen [1] as a means of determining the smallest dimension of ecological phase space. The competition number k(G) of a graph G is defined to be the smallest nonnegative integer k such that G together with k isolated vertices added is the competition graph of an acyclic digraph. We investigate the competition number of a generalized line graph which was introduced by Hoffman [2] in 1970. Subsection 2.1 gives some observations on the competition graphs of acyclic digraphs which will be used in this paper. Subsection 2.2 shows that the competition number of a generalized line graph is at most two. In Subsection 2.3, we investigate generalized line graphs whose competition numbers are one, and give some sufficient conditions and necessary conditions.
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