The competition number of a graph in which any two holes share at most   one edge
  The competition graph of a digraph D is a (simple undirected) graph which has the same vertex set as D and has an edge between x and y if and only if there exists a vertex v in D such that (x,v) and (y,v) are arcs of D. For any graph G, G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number k(G) of G is the smallest number of such isolated vertices. In general, it is hard to compute the competition number k(G) for a graph G and it has been one of important research problems in the study of competition graphs to characterize a graph by its competition number. A hole of a graph is a cycle of length at least 4 as an induced subgraph. It holds that the competition number of a graph cannot exceed one plus the number of its holes if G satisfies a certain condition. In this paper, we show that the competition number of a graph with exactly h holes any two of which share at most one edge is at most h+1, which generalizes the existing results on this subject. 
Jung Yeun Lee, Suh-Ryung Kim, Yoshio Sano
math.CO