Han Hyuk Cho

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.

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.

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.

Domination graphs of regular tournaments

Abstract Let ℘ ∗ (m,n) denote the set of all graphs that are the union of m even paths and n nontrivial odd paths, and ℘(m,n) denote the set of all graphs that are the union of m even paths and n odd paths. In this paper, we show that if G is the domination graph of a regular tournament then G∈℘(m,n) or G is an odd cycle. Also we give a necessary and sufficient condition for G∈℘ ∗ (m,n) to be the domination graph of a regular tournament. Constructions used in this paper will provide insight into the structure of a large class of regular tournaments.

Factorizations of matrices over semirings

In this paper, as a generalization of prime Boolean matrices and prime fuzzy matrices, we study semiprime matrices over chain semirings using the relationship between semiprime matrices and their row spaces. We show that a nonmonomial matrix with full semiring rank can be expressed as a product of elementary matrices and semiprime matrices. Furthermore, we show that a nonmonomial, regular matrix with full semiring rank can be expressed as a product of elementary matrices.

On A(R,S) all of whose members are indecomposable

Let m and n be positive integers, and let R=(r1,r2,…,rm) and S=(s1,s2,…,sn) be non-negative integral vectors with r1+⋯+rm=s1+⋯+sn. In this paper, we present some sufficient conditions for any (0,1)-matrix with row sum vector R and column sum vector S to be indecomposable.