• Journal of applied mathematics & informatics
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• 제5권3호
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• pp.771-784
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• 1998

In this paper we propose an algorithm for generating minimal cutsets of undirected graphs. The algorithm is based on a blocking mechanism for generating every minimal cutest ex-actly once. The algorithm has an advantage of not requiring any preliminary steps to find minimal cutsets. The algorithm generates minimal cutsets at O(e.n) {where e,n = number of (edges, vertices) in the graph} computational effort per cutset. Formal proofs of the algorithm are presented.

• KSII Transactions on Internet and Information Systems (TIIS)
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• 제13권6호
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• pp.3316-3332
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• 2019

In a transitive signature scheme, a signer wants to authenticate edges in a dynamically growing and transitively closed graph. Using transitive signature schemes it is possible to authenticate an edge (i, k), if the signer has already authenticated two edges (i, j) and (j, k). That is, it is possible to make a signature on (i, k) using two signatures on (i, j) and (j, k). We propose the first transitive signature schemes for undirected graphs from lattices. Our first scheme is provably secure in the random oracle model and our second scheme is provably secure in the standard model.

• 경영과학
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• 제26권1호
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• pp.65-76
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• 2009

The undirected Steiner tree problem in graphs is known to be NP-hard. The objective of this problem is to find a shortest tree containing a subset of nodes, called terminal nodes. This paper proposes a method based on a two-step procedure to solve this problem efficiently. In the first step. graph reduction rules eliminate useless nodes and edges which do not contribute to make an optimal solution. In the second step, a max-min ant colony optimization combined with Prim's algorithm is developed to solve the reduced problem. The proposed algorithm is tested in the sets of standard test problems. The results show that the algorithm efficiently presents very correct solutions to the benchmark problems.

• Kyungpook Mathematical Journal
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• 제61권3호
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• pp.661-669
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• 2021

For a simple, undirected graph G = (V, E), a perfect Roman dominating function (PRDF) f : V → {0, 1, 2} has the property that, every vertex u with f(u) = 0 is adjacent to exactly one vertex v for which f(v) = 2. The weight of a PRDF is the sum f(V) = ∑_v∈V f(v). The minimum weight of a PRDF is called the perfect Roman domination number, denoted by γ_R^P(G). Given a graph G and a positive integer k, the PRDF problem is to check whether G has a perfect Roman dominating function of weight at most k. In this paper, we first investigate the complexity of PRDF problem for some subclasses of bipartite graphs namely, star convex bipartite graphs and comb convex bipartite graphs. Then we show that PRDF problem is linear time solvable for bounded tree-width graphs, chain graphs and threshold graphs, a subclass of split graphs.

• 대한수학회지
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• 제52권6호
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• pp.1323-1336
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• 2015

Let S be a semigroup with 0 and R be a ring with 1. We extend the definition of the zero-divisor graphs of commutative semigroups to not necessarily commutative semigroups. We define an annihilating-ideal graph of a ring as a special type of zero-divisor graph of a semigroup. We introduce two ways to define the zero-divisor graphs of semigroups. The first definition gives a directed graph ${\Gamma}$(S), and the other definition yields an undirected graph ${\overline{\Gamma}}$(S). It is shown that ${\Gamma}$(S) is not necessarily connected, but ${\overline{\Gamma}}$(S) is always connected and diam$({\overline{\Gamma}}(S)){\leq}3$. For a ring R define a directed graph ${\mathbb{APOG}}(R)$ to be equal to ${\Gamma}({\mathbb{IPO}}(R))$, where ${\mathbb{IPO}}(R)$ is a semigroup consisting of all products of two one-sided ideals of R, and define an undirected graph ${\overline{\mathbb{APOG}}}(R)$ to be equal to ${\overline{\Gamma}}({\mathbb{IPO}}(R))$. We show that R is an Artinian (resp., Noetherian) ring if and only if ${\mathbb{APOG}}(R)$ has DCC (resp., ACC) on some special subset of its vertices. Also, it is shown that ${\overline{\mathbb{APOG}}}(R)$ is a complete graph if and only if either $(D(R))^2=0,R$ is a direct product of two division rings, or R is a local ring with maximal ideal m such that ${\mathbb{IPO}}(R)=\{0,m,m^2,R\}$. Finally, we investigate the diameter and the girth of square matrix rings over commutative rings $M_{n{\times}n}(R)$ where $n{\geq} 2$.

• 대한수학회지
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• 제52권1호
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• pp.81-96
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• 2015

The intersection graph of a group G is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper non-trivial subgroups of G, and there is an edge between two distinct vertices H and K if and only if $H{\cap}K{\neq}1$ where 1 denotes the trivial subgroup of G. In this paper we characterize all finite groups whose intersection graphs are planar. Our methods are elementary. Among the graphs similar to the intersection graphs, we may count the subgroup lattice and the subgroup graph of a group, each of whose planarity was already considered before in [2, 10, 11, 12].

• 한국경영과학회지
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• 제21권3호
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• pp.215-225
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• 1996

For an undirected stochastic network G, the 2-terminal reliability of G, R(G) is the probability that the specific two nodes (called as terminal nodes) are connected in G. A. typical network reliability problem is to compute R(G). It has been shown that the computation problem of R(G) is NP-hard. So, any algorithm to compute R(G) has a runngin time which is exponential in the size of G. If by some means, the problem size, G is reduced, it can result in immense savings. The means to reduce the size of the problem are the reliability preserving reductions and graph decompositions. We introduce a net set of reliability preserving reductions : the $K^{4}$ (complete graph of 4-nodes)-chain reductions. The total number of the different $K^{4}$ types in R(G), is 6. We present the reduction formula for each $K^{4}$ type. But in computing R(G), it is possible that homeomorphic graphs from $K^{4}$ occur. We devide the homemorphic graphs from $K^{4}$ into 3 types. We develop the reliability preserving reductions for s types, and show that the remaining one is divided into two subgraphs which can be reduced by $K^{4}$-chain reductions 7 polygon-to-chain reductions.

• 한국컴퓨터정보학회논문지
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• 제19권7호
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• pp.103-111
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• 2014

본 논문은 실시간 GPS 항법시스템에서 최단경로 탐색에 일반적으로 적용되고 있는 Dijkstra 알고리즘을 양방향 통행로(무방향그래프)로만 구성된 도로에 적용하고 문제점을 개선한 알고리즘을 제안하였다. Dijkstra 알고리즘은 방향 그래프에서 출발 노드부터 시작하여 그래프의 모든 노드에 대한 최단경로를 결정하기 때문에 알고리즘 수행에 많은 메모리가 요구되어 실시간으로 정보를 제공하지 못할 수도 있다. 이러한 문제점을 해결하고자, 본 논문에서는 무방향 그래프에 적합하도록 출발과 목적지 정점을 제외한 경로 정점들에 대해 최단경로를 설정하고, 출발 정점부터 시작하여 정점 유출 간선들에 대해 최단경로 설정 간선들과 일치하는 간선들을 모두 선택하는 방식으로 한 번에 다수의 정점들을 탐색하는 방법을 택하였다. 9개의 다양한 무방향 그래프에 제안된 알고리즘을 적용한 결과 모두 최단경로를 탐색하는데 성공하였다. 또한, 수행 속도 측면에서 Dijkstra 알고리즘보다 약 60%를 단축시키는 효과를 얻었으며, 알고리즘 수행에 필요한 메모리도 월등히 적게 요구되었다.

• 호남수학학술지
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• 제40권1호
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• pp.75-82
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• 2018

Let P be a commutator poset with Z(P) its set of zero-divisors. The annihilator graph of P, denoted by AG(P), is the (undirected) graph with all elements of $Z(P){\setminus}\{0\}$ as vertices, and distinct vertices x, y are adjacent if and only if $ann(xy)\;{\neq}\;(x)\;{\cup}\;ann(y)$. In this paper, we study basic properties of AG(P).

• 대한수학회논문집
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• 제18권4호
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• pp.745-754
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• 2003

Maximum clique problem is to find a maximum clique(largest in size) in an undirected graph G. We present a method that computes either a maximum clique or an upper bound for the size of a maximum clique in G. We show that this method performs well on certain class of graphs and discuss the application of this method in a branch and bound algorithm for solving maximum clique problem, whose efficiency is depended on the computation of good upper bounds.