Suh-Ryung Kim

The Competition Number and its Variants

Abstract If D is an acyclic digraph, its competition graph has the same vertex set as D and an edge between vertices u and v if and only if for some vertex w of D, there are arcs (u,w) and (v,w) in D. The competition number of a graph G is the smallest number of isolated vertices whose addition makes G into a competition graph. Competition graphs were introduced by Cohen in 1968 as a means of determining the smallest dimension of ecological phase space. Various notions analogous to competition graphs together with competition graphs have applications, not only to ecology, but in studying communication over a noisy channel, in assigning frequencies to radio transmitters, and in modeling complex economic and energy systems. In the study of the competition graph of an acyclic digraph, there are two fundamental questions which were proposed by Roberts in 1978: One is to characterize the acyclic digraphs which have interval competition graphs and the other is to characterize the graphs which are competition graphs of acyclic digraphs. In this paper, we focus our interest on the second question as we survey the results about the competition number and some of its variants, namely, the p-competition number, the double competition number, and the niche number. Open questions related to the topic are discussed as well.

The m -step competition graph of a digraph

The competition graph of a digraph was introduced by Cohen in 1968 associated with the study of ecosystems. Since then, the competition graph has been widely studied and many variations have been introduced. In this paper, we define and study the m-step competition graph of a digraph which is another generalization of competition graph.

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.

The competition numbers of complete tripartite graphs

For a graph G, it is known to be a hard problem to compute the competition number k(G) of the graph G in general. In this paper, we give an explicit formula for the competition numbers of complete tripartite graphs.

The competition number of a graph having exactly one hole

Let D be an acyclic digraph. The competition graph of D has the same set of vertices as D and an edge between vertices u and v if and only if there is a vertex x in D such that (u,x) and (v,x) are arcs of D. The competition number of a graph G, denoted by k(G), is the smallest number k such that G together with k isolated vertices is the competition graph of an acyclic digraph. In this paper, we show that the competition number of a graph having exactly one chordless cycle of length at least 4 is at most two. We also give a large family of such graphs whose competition numbers are less than or equal to one.

p-competition graphs

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 and arcs ( x , a i ) and ( y , a i ) in D for each i ≤ p . This definition generalizes the widely studied p = 1 case of ordinary competition graphs. We obtain results about p -competition graphs analogous to the well-known results about ordinary competition graphs and apply these results to specific cases.

2-competition graphs

IfD (V, A) is a digraph, its p-competition graph for p a positive integer has vertex set V and an edge between x and y ifand only if there are distinct vertices a, , an in D with (x, a and (y, a) arcs ofD for each 1, , p. This notion generalizes the notion of ordinary competition graph, which has been widely studied and is the special case wherep 1. Results about the case wherep 2 are obtained. In particular, the paper addresses the question ofwhich complete bipartite graphs are 2-competition graphs. This problem is formulated as the following combinatorial problem: Given disjoint setsA and B such that A tO BI n, when can one find n subsets ofA tO B so that every a in A and b in B are together contained in at least two of the subsets and so that the intersection of every pair of subsets contains at most one element from A and at most one element from B?

A class of acyclic digraphs with interval competition graphs

Let D be an acyclic digraph. The competition graph of D has the same set of vertices as D and an edge between vertices u and v if and only if there is a vertex x in D such that (u, x) and (v, x) are arcs of D. In this paper, we show that the competition graphs of doubly partial orders are interval graphs. We also show that an interval graph together with enough isolated vertices is the competition graph of a doubly partial order. Finally, we introduce a new notion called the doubly partial order competition number of an interval graph and present some open questions.

The competition numbers of complete multipartite graphs and mutually orthogonal Latin squares

The competition graph of a digraph D is the graph which has the same vertex set as D and has an edge between u and v if and only if there exists a vertex x in D such that (u,x) and (v,x) are arcs of D. For any graph G, the disjoint union of G and sufficiently many isolated vertices is the competition graph of some acyclic digraph. The smallest number of isolated vertices needed is defined to be the competition number k(G) of G. In general, it is hard to compute the competition number k(G) for a graph G and it is an important research problem in the study of competition graphs to characterize the competition graphs of acyclic digraphs by computing their competition numbers. In this paper, we give new upper and lower bounds for the competition number of a complete multipartite graph in which all partite sets have the same size by using orthogonal Latin squares. When there are exactly four partite sets, we give better bounds.

The competition number of a graph whose holes do not overlap much

Let D be an acyclic digraph. The competition graph of D is a graph which has the same vertex set as D and has an edge between u and v if and only if there exists a vertex x in D such that (u,x) and (v,x) 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. A hole of a graph is an induced cycle of length at least four. Kim (2005) [8] conjectured that the competition number of a graph with h holes is at most h+1. Recently, Li and Chang (2009) [11] showed that the conjecture is true when the holes are independent. In this paper, we show that the conjecture is true though the holes are not independent but mutually edge-disjoint.