Competition Numbers of Graphs with a Small Number of Triangles

If D is an acyclic digraph, its competition graph is an undirected graph with the same vertex set and an edge between vertices x and y if there is a vertex a so that (x, a) and (y, a) are both arcs of D. If G is any graph, G together with sufficiently many isolated vertices is a competition graph, and the competition number of G is the smallest number of such isolated vertices. Roberts (1978) gives a formula for the competition number of connected graphs with no triangles. In this paper, we compute the competition numbers of connected graphs with exactly one or exactly two triangles.