• 대한수학회보
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• 제37권4호
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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.

• 대한수학회지
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• 제42권6호
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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[3] 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.

• 한국안전학회지
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• 제11권1호
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• pp.27-38
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• 1996

현대 산업사회에서 네트워크나 시스템의 신뢰도는 전기, 전자, 통신, 화공, 핵공학 등에서 광범위하게 사용되고 있다. 통신의 경우 순간의 고장도 사회에 직접적으로 미치는 파급 효과가 크므로, 예측되는 부품의 고장 둥으로 인한 네트워크의 사용 가능 여부, 확률 등이 관심의 대상이 되고 있다. 이러한 문제들은 종종 신뢰도 공학의 고전적인 문제인 source to terminal problem으로 표현될 수 있으며, 이들의 문제해결을 위하여 그래프 이론과 domination 이론이 점점 중요한 비중을 차지하고 있다. 예를 들면 13개의 minimal path로 구성된 어떤 네트워크(그레프)를 관찰할 때, 신뢰도 계산을 위하여는 이들 13개 m-path의 모든 조합($2^{l3}$ -1=8191개)을 관찰하여야 하나, $\ulcorner$l$\lrcorner$에서 발표된 예제는 domination 이론을 기초로 한 topologic식을 사용하면 정확성의 상실없이 123개 항으로 감소시킬 수 있음을 보여주었다. A. Satyanarayana와 A. Prabhaker등은 $\ulcorner$1-19$\lrcorner$에서 그래프로 표현되는 시스템이나 네트워크들의 정확한 신뢰도 계산을 위하여 m-path를 사용한 domination 이론을 연구하고, 몇가지 알고리즘을 제시하였다. 하지만 어떤 네트워크를 관찰하 때 "왜 정상인가\ulcorner" 보다는 "왜 고장인가\ulcorner"를 관찰하여야 할 경우가 더 많으며, 이런 경우 m-path보다는 m-cutset을 사용한 신뢰도분석이 더 요구된다. $\ulcorner$20$\lrcorner$에서는 m-cutset을 근거로 한 네트워크(그래프)의 domination을 연구하였으나, $\ulcorner$1$\lrcorner$dml m-path를 기초로 한 경우처럼 간단한 topologic식이 성립 될 수 없음을 밝혔다.(중략)

• 대한수학회지
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• 제48권4호
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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.

• 대한수학회지
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• 제59권5호
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• pp.963-986
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• 2022

Given a dimension function 𝜔, we introduce the notion of an 𝜔-vector weighted digraph and an 𝜔-equivalence between them. Then we establish a bijection between the weakly (ℤ/2)ⁿ-equivariant homeomorphism classes of small covers over a product of simplices ∆^𝜔(1) × ⋯ × ∆^𝜔(m) and the set of 𝜔-equivalence classes of 𝜔-vector weighted digraphs with m-labeled vertices, where n is the sum of the dimensions of the simplicies. Using this bijection, we obtain a formula for the number of weakly (ℤ/2)ⁿ-equivariant homeomorphism classes of small covers over a product of three simplices.