P-competition numbers

Abstract

If D = ( V , A >) is a digraph, its p -competition graph has vertex set V and an edge between x and y if and only if there are distinct vertices a 1 ,…, a p in D with ( x, a i ) and ( y,a i ) arcs of D for each i ⩽ p . The p -competition number of a graph is the smallest number of isolated vertices which need to be added in order to make it a p -competition graph. These notions generalize the widely studied p = 1 case, where they correspond to ordinary competition graphs and competition numbers. We obtain bounds on the p -competition number in terms of the ordinary competition number, and show that, surprisingly, the p -competition number can be arbitrarily smaller than the ordinary competition number.