• Title, Summary, Keyword: acyclic digraph

### ACYCLIC DIGRAPHS WHOSE 2-STEP COMPETITION GRAPHS ARE P$P_n\cup\ I_2$

• Cho, Han-Hyun;Kim, Suh-Ryung;Nam, Yunsun
• Bulletin of the Korean Mathematical Society
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• v.37 no.4
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• pp.649-657
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• 2000
• The 2-step competition graph of D has the same vertex set as D and an edge between vertices x and y if and only if there exist (x, z)-walk of length 2 and (y, z)-walk of length 2 for some vertex z in D. The 2-step competition number of a graph G is the smallest number k such that G together with k isolated vertices is the 2-step competition graph of an acyclic digraph. Cho, et al. showed that the 2-step competition number of a path of length at least two is two. In this paper, we characterize all the minimal acyclic digraphs whose 2-step competition graphs are paths of length n with two isolated vertices and construct all such digraphs.

### GRAPHS WITH ONE HOLE AND COMPETITION NUMBER ONE

• KIM SUH-RYUNG
• Journal of the Korean Mathematical Society
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• v.42 no.6
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• pp.1251-1264
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• 2005
• 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. It is known to be difficult to compute the competition number of a graph in general. Even characterizing the graphs with competition number one looks hard. In this paper, we continue the work done by Cho and Kim to characterize the graphs with one hole and competition number one. We give a sufficient condition for a graph with one hole to have competition number one. This generates a huge class of graphs with one hole and competition number one. Then we completely characterize the graphs with one hole and competition number one that do not have a vertex adjacent to all the vertices of the hole. Also we show that deleting pendant vertices from a connected graph does not change the competition number of the original graph as long as the resulting graph is not trivial, and this allows us to construct infinitely many graph having the same competition number. Finally we pose an interesting open problem.

### A study of fast Reliability evaluation for acyclic digraph using domination theory (Domination 이론을 이용한 acyclic digraph의 빠른 신뢰도 계산을 위한 연구)

• 이광원
• Journal of the Korean Society of Safety
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• v.11 no.1
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• pp.27-38
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• 1996
• The aim of this paper is to develop more fast algorithm for evaluation of the reliability of networks and system. It is illustrated with examples. This paper derived the algorithm to calculate the acyclic directed graph G(deals with the problem of the s-t graph). The language PASCAL was used to implement the algorithm. Three Examples are calculated and the calculation time is shorter than the time by program in $\ulcorner$21\$\lrcorner.

### THE COMPETITION NUMBERS OF HAMMING GRAPHS WITH DIAMETER AT MOST THREE

• Park, Bo-Ram;Sano, Yoshio
• Journal of the Korean Mathematical Society
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• v.48 no.4
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• pp.691-702
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• 2011
• The competition graph of a digraph D is a 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 a graph G is defined to be 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. In this paper, we compute the competition numbers of Hamming graphs with diameter at most three.