• East Asian mathematical journal
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• 제31권5호
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• pp.717-725
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• 2015

Velzen introduced the rigid and relaxed dominating set games and showed that the rigid game being balanced is equivalent to the relaxed game being balanced in 2004. After then various variants of dominating set games were introduced and it was shown that for each variant, a rigid game being balanced is equivalent to a relaxed game being balanced. It is natural to ask if for any other variant of dominating set game, the balancedness of a rigid game and the balancedness of a relaxed game are equivalent. In this paper, we show that the answer for the question is negative by considering the rigid and relaxed paired-domination games, which is considered as a variant of dominating set games. We characterize the cores of both games and show that the rigid game being balanced is not equivalent to the relaxed game being balanced. In addition, we study the cores of paired-dominations games on paths and cycles.

• KSII Transactions on Internet and Information Systems (TIIS)
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• 제10권4호
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• pp.1661-1678
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• 2016

Clustering is one of the key technologies in vehicular networks. Constructing and maintaining stable clusters is a challenging task in high mobility environments. DWCM (Distributed and Weighted Clustering based on Mobility Metrics) is proposed in this paper based on the d-hop dominating set of the network. Each vehicle is assigned a priority that describes the cluster relationship. The cluster structure is determined according to the d-hop dominating set, where the vehicles in the d-hop dominating set act as the cluster head nodes. In addition, cluster maintenance handles the cluster structure changes caused by node mobility. The rationality of the proposed algorithm is proven. Simulation results in the NS-2 and VanetMobiSim integrated environment demonstrate the performance advantages.

• Journal of information and communication convergence engineering
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• 제9권1호
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• pp.105-109
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• 2011

In this paper, we study the dominating-free sest which is defined as follows: k points called servers and n points called clients in the plane are given. For a point p in the plane is said to be dominated by a client c if for every server s, the distance between s and p is greater than the distance between s and c. The dominating-free set is the set of points in the plane which aren't dominated by any client. We present an O(nklogk+$n^2k$) time algorithm for computing the dominating-free set under the $L_1$-metric. Specially, we present an O(nlogn) time algorithm for the problem when k=2. The algorithm uses some variables and 1-dimensional arrays as its principle data structures, so it is easy to implement and runs fast.

• Journal of applied mathematics & informatics
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• 제41권4호
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• pp.741-752
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• 2023

A new variant of vertex edge domination, namely semi total vertex edge domination has been introduced in the present paper. A subset S of the vertex set V of a graph G is said to be a semi total vertex edge dominating set(stved - set), if it is a vertex edge dominating set of G and each vertex in S is within a distance two of another vertex in S. An stved-set of G having minimum cardinality is said to be an γ_stve(G)- set and its cardinality is denoted by γ_stve(G). Bounds for γ_stve(G) - set have been given in terms of various graph theoretic parameters and graphs attaining the bounds have been characterized. In particular, bounds for trees have been obtained and extremal trees have been characterized.

• Korean Journal of Mathematics
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• 제29권1호
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• pp.1-12
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• 2021

A dominating set S of a graph G is said to be nonsplit dominating set if the subgraph ⟨V - S⟩ is connected. The minimum cardinality of a nonsplit dominating set is called the nonsplit domination number and is denoted by ��ns(G). For a minimum nonsplit dominating set S of G, a set T ⊆ S is called a forcing subset for S if S is the unique ��ns-set containing T. A forcing subset for S of minimum cardinality is a minimum forcing subset of S. The forcing nonsplit domination number of S, denoted by f��ns(S), is the cardinality of a minimum forcing subset of S. The forcing nonsplit domination number of G, denoted by f��ns(G) is defined by f��ns(G) = min{f��ns(S)}, where the minimum is taken over all ��ns-sets S in G. The forcing nonsplit domination number of certain standard graphs are determined. It is shown that, for every pair of positive integers a and b with 0 ≤ a ≤ b and b ≥ 1, there exists a connected graph G such that f��ns(G) = a and ��ns(G) = b. It is shown that, for every integer a ≥ 0, there exists a connected graph G with f��(G) = f��ns(G) = a, where f��(G) is the forcing domination number of the graph. Also, it is shown that, for every pair a, b of integers with a ≥ 0 and b ≥ 0 there exists a connected graph G such that f��(G) = a and f��ns(G) = b.

• 대한수학회논문집
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• 제18권1호
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• pp.181-192
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• 2003

We count the numbers of independent dominating sets of rooted labeled trees, ordinary labeled trees, and recursive trees, respectively.

• 한국정보과학회논문지:시스템및이론
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• 제27권6호
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• pp.608-618
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• 2000

본 논문에서는 병렬 및 분산 시스템에서 자원을 배치함에 있어서 최소한의 자원 복사(copy)만을 사용하면서 임의의 노드 및 링크 상에서 고장이 발생하더라도 주어진 성능 요건을 만족하게 하는 자원의 최적 배치 방법을 모색하고자 한다. 이러한 성능 요건의 만족과 시스템의 고가용성을 위하여, 모든 노드들에 대하여 최소한의 자원 복사를 사용하여 그 노드나 혹은 인접한 노드 중 적어도 두 개 이상에 자원 복사가 존재해야 하는데, 이것을 본 논문에서는 고장 허용 자원 배치 문제라고 부른다. 병렬 및 분산 시스템은 그래프로 표현할 수가 있다. 여기에서 고장 허용 자원 배치 문제는 그래프 상에서 가장 작은 고장 허용 dominating set을 찾는 문제로 변환이 된다. Dominating set 문제는 NP-complete로 증명이 되어 있으며, 본 논문에서는 A* 알고리즘을 사용하여 상태 공간 탐색 방법으로 최적 배치를 구한다. 또한, 최적 배치를 찾는 데에 걸리는 시간을 단축시키기 위하여, 고장 허용 dominating set의 특성들을 분석하여 유용한 휴리스틱 정보들을 도출한다. 또한 여러가지 정형 그래프와 임의 그래프 상에서의 실험을 통하여, 이들 휴리스틱 정보들을 사용하여 최적 고장 허용 자원 배치를 찾는 데에 걸리는 시간을 상당히 줄일 수 있음을 보인다.

• Korean Journal of Mathematics
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• 제31권4호
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• pp.505-519
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• 2023

Let G = (V, E) be a graph. A subset S of V is called a dominating set of G if every vertex not in S is adjacent to some vertex in S. The domination number γ(G) of G is the minimum cardinality taken over all dominating sets of G. A dominating set S is called a connected dominating set if the subgraph induced by S is connected. The minimum cardinality taken over all connected dominating sets of G is called the connected domination number of G, and is denoted by γ_c(G). In this paper, we investigate the Nordhaus-Gaddum type results for the connected domination number and its derived graphs like line graph, subdivision graph, power graph, block graph and total graph, and characterize the extremal graphs.

• Kyungpook Mathematical Journal
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• 제56권1호
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• pp.69-81
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• 2016

For a positive integer k, a k-rainbow dominating function of a digraph D is a function f from the vertex set V (D) to the set of all subsets of the set $\{1,2,{\ldots},k\}$ such that for any vertex $v{\in}V(D)$ with $f(v)={\emptyset}$ the condition ${\cup}_{u{\in}N^-(v)}$ $f(u)=\{1,2,{\ldots},k\}$ is fulfilled, where $N^-(v)$ is the set of in-neighbors of v. A set $\{f_1,f_2,{\ldots},f_d\}$ of k-rainbow dominating functions on D with the property that $\sum_{i=1}^{d}{\mid}f_i(v){\mid}{\leq}k$ for each $v{\in}V(D)$, is called a k-rainbow dominating family (of functions) on D. The maximum number of functions in a k-rainbow dominating family on D is the k-rainbow domatic number of D, denoted by $d_{rk}(D)$. In this paper we initiate the study of the k-rainbow domatic number in digraphs, and we present some bounds for $d_{rk}(D)$.

• 한국통신학회논문지
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• 제35권12A호
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• pp.1188-1197
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• 2010

무선 애드혹 망에서 connected dominating set(CDS)를 활용한 라우팅 방식의 핵심은 dominating 노드로 동작할 최소 개수의 노드들을 선택하고, 이 노드들로 이루어진 백본 망을 구성하는 것이다. CDS 에서 장애 노드가 발생할 확률은 무시할 수 있는 수준은 아니다. 고장 감내가 중요한 비중을 차지하는 응용에서는 기존 CDS 기반 라우팅이 바람직하지 않을 수 있다. 따라서 메시지 플러딩에 따른 오버헤드로 인해 CDS 전체 재구성 시도를 최소화하는 것이 필요하다. 이를 위해 CDS 전체 재구성을 시도하는 대신, 장애가 발생한 노드를 중심으로 제한된 범위에 놓인 노드들에 대해서만 CDS를 부분 재구성할 수 있도록 대체 노드를 찾는 방안을 제안한다. 이러한 방식을 적용할 경우., CDS 부분 재구성시에도 dominating 노드 수가 전체 재구성을 시도했을 때와 같게 유지될 뿐만 아니라 전체 재구성 때보다 20~40% CDS 구성 시간을 단축시킬 수 있다. 고 이동성을 갖는 상황에서 기존 전체 재구성 알고리즘에 비해 패킷 수신율 및 에너지 소비 측면에서 유리한 결과를 얻었다.