Fe b 20 16 A study on competition numbers of graphs in the aspect of primary predator index ∗

The competition graph of a digraph D is defined as a graph which has the same vertex set as D and has an edge xy between two distinct vertices x and y if and only if, for some vertex z ∈ V (D), the arcs (x, z) and (y, z) are in D. The competition number of a graph G is defined to be the smallest number k such that G together with k isolated vertices is the competition graph of an acyclic digraph. In this paper, we define the primary predator index p of a graph G with competition number k as the largest number p such that an acyclic digraph whose competition graph is G together with k isolated vertices has p vertices of in-degree 0, and show that the competition number k(G) and the primary predator index p(G) of a graph G are related as k(G) ≥ θe(G) − |V (G)| + p(G). As a matter of fact, this generalizes the inequality k(G) ≥ θe(G)−|V (G)|+2 given by Opsut [7] as p(G) ≥ 2 for a graph with at least one edge. In the process, we introduce a notion of effective competition cover, which is possessed by a large family of graphs, and show that our inequality actually becomes an equality for a graph having an effective competition cover and the competition number of a certain graph having an effective competition cover may be obtained by utilizing its primary predator index. Especially, we present a precise relationship between the competition number and the primary predator index of a diamond-free plane graph. This work forms part of the author’s master’s thesis at Seoul National University. This research was supported by Global Ph.D Fellowship Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (No. NRF-2015H1A2A1033541). Corresponding author: [email protected] This work was partially supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. NRF-2015R1A2A2A01006885).