• Journal of applied mathematics & informatics
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• 제32권1_2호
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• pp.1-6
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• 2014

Let G = (V ; E) be a simple undirected graph without loops and multiple edges, the vertex and edge sets of it are represented by V = V (G) and E = E(G), respectively. A weighting w of the edges of a graph G induces a coloring of the vertices of G where the color of vertex v, denoted $S_v:={\Sigma}_{e{\ni}v}\;w(e)$. A k-edge-weighting of a graph G is an assignment of an integer weight, w(e) ${\in}${1,2,...,k} to each edge e, such that two vertex-color $S_v$, $S_u$ be distinct for every edge uv. In this paper we determine an exact 3-edge-weighting of complete graphs $k_{3q+1}\;{\forall}_q\;{\in}\;{\mathbb{N}}$. Several open questions are also included.

• 한국수학교육학회지시리즈A:수학교육
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• 제13권2호
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• pp.6-10
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• 1974

We consider 'ordinary' graphs: that is, finite undirected graphs with no loops or multiple edges. An intersection representation of a graph G is a function r from V(G), the set of vertices of G, into a family of sets S such that distinct points $\chi$$_{\alpha}$ and $\chi$$_{\beta}$/ of V(G) are. neighbors in G precisely when ${\gamma}$($\chi$$_{\alpha}$)∩${\gamma}$($\chi$$_{\beta}$/)$\neq$ø, A graph G is a rigid circuit grouph if every cycle in G has at least one triangular chord in G. In this paper we consider the main theorem; A graph G has an intersection representation by arcs on an acyclic graph if and only if is a normal rigid circuit graph.uit graph.d circuit graph.uit graph.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제11권3호
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• pp.65-75
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• 2007

A graph G is self-complementary (sc) if it is isomorphic to its complement. G is perfect if for all induced subgraphs H of G, the chromatic number of H (denoted ${\chi}$(H)) equals the number of vertices in the largest clique in H (denoted ${\omega}$(H)). An sc graph which is also perfect is known as sc perfect graph. A comparability graph is an undirected graph if it can be oriented into transitive directed graph. An sc comparability (scc) is clearly a subclass of sc perfect graph. In this paper we show that no two non-isomorphic scc graphs with n vertices each, (n<13) have same spectrum, and that the smallest positive integer for which there exists hyper-energetic scc graph is 13.

• Kyungpook Mathematical Journal
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• 제62권1호
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• pp.193-204
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• 2022

A subset S of V(G), where G is a simple undirected graph, is a hop dominating set if for each v ∈ V(G)＼S, there exists w ∈ S such that d_G(v, w) = 2 and it is a locating-hop set if N_G(v, 2) ∩ S ≠ N_G(v, 2) ∩ S for any two distinct vertices u, v ∈ V(G)＼S. A set S ⊆ V(G) is a locating-hop dominating set if it is both a locating-hop and a hop dominating set of G. The minimum cardinality of a locating-hop dominating set of G, denoted by 𝛄_lh(G), is called the locating-hop domination number of G. In this paper, we investigate some properties of this newly defined parameter. In particular, we characterize the locating-hop dominating sets in graphs under some binary operations.

• 정보처리학회논문지B
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• 제15B권6호
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• pp.609-612
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• 2008

G = (V, E) 를 단순 무방향성 그래프라 하자. Minimum Vertex Cover (MVC) 문제는 C 를 V 의 부분 집합이라 할 때 모든 간선들이 C 내의 최소 한 개 정점과 인접하게 되는 최소 집합 C 를 계산하는 것이다. 다른 많은 그래프 이론 문제와 마찬가지로 본 문제도 NP-hard 문제임이 증명되었다. 본 논문에서는 MVC 문제를 위한 LeafGA 라는 새로운 유전 알고리즘을 제시하며 또한 제시된 알고리즘을 널리 알려 진 기준 그래프들에 적용함으로써 그 효용성을 보인다.

• 한국인터넷방송통신학회논문지
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• 제22권6호
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• pp.91-97
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• 2022

사이클 검출 문제에 대해, 단일 출발(SS)을 갖는 단일 연결 리스트(SLL)에 한해 O(n) 복잡도의 거북이와 토끼 경주법(HTA)이 제안되었으며, 다중 출발지-다중 종착지, 다중 분기(MSMDMB)를 갖는 일반 그래프에 대해서는 빠른 방법이 알려져 있지 않고 있다. 본 논문에서는 MSMDMB를 갖는 주어진 무 방향과 방향 그래프의 최대 사이클을 선형시간 복잡도로 검출할 수 있는 방법을 제안하였다. 제안된 방법은 주어진 원 그래프 G에는 사이클 형성 조건을 충족시키지 못하는 다수의 정점(또는 노드)가 존재한다는 사실에 기반하여 이들 정점(또는 노드)들을 제거한 축소된 그래프 G'를 얻었다. 이 축소된 그래프에 대해 선형시간 복잡도인 선형탐색으로 사이클 집합 C와 사이클 길이 λ를 찾았다. 제안된 알고리즘을 실험 데이터에 적용한 결과 모든 데이터들에 대해 최대 사이클을 찾을 수 있음을 보였다.

• 한국인터넷방송통신학회논문지
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• 제12권3호
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• pp.51-61
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• 2012

본 논문에서는 무방향 그래프의 최소신장트리 (Minimum Spanning Tree, MST) 알고리즘인 Prim MST 알고리즘으로 방향 그래프의 최소신장트리 (DMST)를 구하는 알고리즘을 제안하였다. 먼저, 무방향 그래프와 방향 그래프의 차이점을 반영하여 각 노드에서 유출되는 호들 중 최소 가중치를 가진 호 (Minimum Weight Arc, MWA)를 선택하는 Prim DMST 알고리즘을 제안하였다. 다음으로 Prim DMST 알고리즘과 DMST의 대표적인 Chu-Liu/Edmonds DMST 알고리즘을 실제 3개 그래프에 적용하여 DMST를 찾지 못하는 단점을 보였다. 마지막으로 항상 DMST를 찾을 수 있는 알고리즘으로 Prim DMST를 변형시킨 진보된 Prim DMST 알고리즘을 제안하였다. Prim DMST 알고리즘은 각 노드의 유출 호들 중 MWA를 선택하는 방법이다. 반면에 진보된 Prim DMST 알고리즘은 각 노드의 유출 호들과 유입 호들 중 일치하는 호들을 선택하는 방법을 택하였으며, 만약에 일치하는 호가 없을 경우 각 노드의 유출 호들 중 MWA를 선택하는 방법이다. 제안된 알고리즘을 17개의 다양한 그래프에 적용한 결과, 항상 Chu-Liu/Edmonds DMST 알고리즘과 동일한 DMST를 찾는데 성공하였다. 또한, Chu-Liu/Edmonds DMST 알고리즘과 같이 사이클을 제거하기 위한 복잡한 계산을 하지 않아도 되며, Prim DMST 알고리즘 보다 수행속도를 크게 단축시킬 수 있었다.

• 한국생물정보학회:학술대회논문집
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• 한국생물정보시스템생물학회 2005년도 BIOINFO 2005
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• pp.267-271
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• 2005

The goal of data mining is to extract new and useful knowledge from large scale datasets. As the amount of available data grows explosively, it became vitally important to develop faster data mining algorithms for various types of data. Recently, an interest in developing data mining algorithms that operate on graphs has been increased. Especially, mining frequent patterns from structured data such as graphs has been concerned by many research groups. A graph is a highly adaptable representation scheme that used in many domains including chemistry, bioinformatics and physics. For example, the chemical structure of a given substance can be modelled by an undirected labelled graph in which each node corresponds to an atom and each edge corresponds to a chemical bond between atoms. Internet can also be modelled as a directed graph in which each node corresponds to an web site and each edge corresponds to a hypertext link between web sites. Notably in bioinformatics area, various kinds of newly discovered data such as gene regulation networks or protein interaction networks could be modelled as graphs. There have been a number of attempts to find useful knowledge from these graph structured data. One of the most powerful analysis tool for graph structured data is frequent subgraph analysis. Recurring patterns in graph data can provide incomparable insights into that graph data. However, to find recurring subgraphs is extremely expensive in computational side. At the core of the problem, there are two computationally challenging problems. 1) Subgraph isomorphism and 2) Enumeration of subgraphs. Problems related to the former are subgraph isomorphism problem (Is graph A contains graph B?) and graph isomorphism problem(Are two graphs A and B the same or not?). Even these simplified versions of the subgraph mining problem are known to be NP-complete or Polymorphism-complete and no polynomial time algorithm has been existed so far. The later is also a difficult problem. We should generate all of 2$^n$ subgraphs if there is no constraint where n is the number of vertices of the input graph. In order to find frequent subgraphs from larger graph database, it is essential to give appropriate constraint to the subgraphs to find. Most of the current approaches are focus on the frequencies of a subgraph: the higher the frequency of a graph is, the more attentions should be given to that graph. Recently, several algorithms which use level by level approaches to find frequent subgraphs have been developed. Some of the recently emerging applications suggest that other constraints such as connectivity also could be useful in mining subgraphs : more strongly connected parts of a graph are more informative. If we restrict the set of subgraphs to mine to more strongly connected parts, its computational complexity could be decreased significantly. In this paper, we present an efficient algorithm to mine frequent subgraphs that are more strongly connected. Experimental study shows that the algorithm is scaling to larger graphs which have more than ten thousand vertices.

• 디지털융복합연구
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• 제19권10호
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• pp.403-410
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• 2021

k-club은 소셜 네트워크 분석에서 다양한 형태의 소셜 그룹을 설명하기 위해 제안된 그래프 모델 중 하나로, 단순 그래프에서 부분 정점 집합 S 에 의한 유도 부분그래프(Induced subgraph)의 지름이 k보다 작거나 같은 경우 S 를 k-club이라 한다. 본 논문에서는 유전알고리즘을 이용하여 그래프에서 크기가 최대인 k-club을 찾는 문제인 MkCP(Maximum k-Club Problem)을 계산하는 HGA+DROP 알고리즘을 제안한다. 본 알고리즘은 k-club을 위한 휴리스틱 알고리즘 k-CLIQUE & DROP을 변형하고 휴리스틱 유전 알고리즘(HGA)을 사용해 한 번의 수행으로 복수개의 k-club을 구하였다. 기존 알고리즘의 결과와 비교하기 위해 DIMACS 그래프들에 대하여 k가 2, 3, 4 그리고 5일 때 MkCP를 계산하였다.

• Kyungpook Mathematical Journal
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• 제62권3호
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• pp.437-453
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• 2022

Unlike for polynomial rings, the notion of multiplication for the near-ring of polynomials is the substitution operation. This leads to somewhat surprising results. Let S be an abelian left near-ring with identity. The relation ~ on S defined by letting a ~ b if and only if ann_S(a) = ann_S(b), is an equivalence relation. The compressed zero-divisor graph 𝚪_E(S) of S is the undirected graph whose vertices are the equivalence classes induced by ~ on S other than [0]_S and [1]_S, in which two distinct vertices [a]_S and [b]_S are adjacent if and only if ab = 0 or ba = 0. In this paper, we are interested in studying the compressed zero-divisor graphs of the zero-symmetric near-ring of polynomials R₀[x] and the near-ring of the power series R₀[[x]] over a commutative ring R. Also, we give a complete characterization of the diameter of these two graphs. It is natural to try to find the relationship between diam(𝚪_E(R₀[x])) and diam(𝚪_E(R₀[[x]])). As a corollary, it is shown that for a reduced ring R, diam(𝚪_E(R)) ≤ diam(𝚪_E(R₀[x])) ≤ diam(𝚪_E(R₀[[x]])).