Two families of graphs that are not CCE-orientable

Abstract

Let D be a digraph. The competition-common enemy graph of D has the same set of vertices as D and an edge between vertices u and v if and only if there are vertices w and x in D such that (w, u), (w, v), (u, x), and (v, x) are arcs of D. We call a graph a CCE-graph if it is the competition-common enemy graph of some digraph. We also call a graph G = (V, E) CCE-orientable if we can give an orientation F of G so that whenever (w, u), (w, v), (u, x), and (v, x) are in F, either (u, v) or (v, u) is in F. Bak et al. [1997] found a large class of graphs that are CCE-orientable and proposed an open question of finding graphs that are not CCE-orientable. In this paper, we answer their question by presenting two families of graphs that are not CCE-orientable. We also give a CCE-graph that is not CCE-orientable, which answers another question proposed by Bak et al. [1997]. Finally we find a new family of graphs that are CCE-orientable.