The k-domatic number of a graph

Kämmerling, Karsten; Volkmann, Lutz

doi:10.1007/s10587-009-0036-0

Cited by 7 publications

(3 citation statements)

References 6 publications

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“…In this section we determine the exact values of the k-domatic number of some special classes of graphs. By Theorem 2.9 in [15], d k (G) ≤ δ (G) k + 1. As d k (G) is a positive integer, we have d k (G) = 1 whenever k > δ(G).…”

Section: K-domatic Number Of Special Graphsmentioning

confidence: 90%

The Algorithmic Complexity of k-Domatic Partition of Graphs

Liang

2012

Lecture Notes in Computer Science

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Let G = (V, E) be a simple undirected graph, and k be a positive integer. A k-dominating set of G is a set of vertices S ⊆ V satisfying that every vertex in V \ S is adjacent to at least k vertices in S. A k-domatic partition of G is a partition of V into k-dominating sets. The k-domatic number of G is the maximum number of k-dominating sets contained in a k-domatic partition of G. In this paper we study the k-domatic number from both algorithmic complexity and graph theoretic points of view. We prove that it is N P-complete to decide whether the k-domatic number of a bipartite graph is at least 3, and present a polynomial time algorithm that approximates the k-domatic number of a graph of order n within a factor of (1 k + o(1)) ln n, generalizing the (1 + o(1)) ln n approximation for the 1-domatic number given in [5]. In addition, we determine the exact values of the k-domatic number of some particular classes of graphs.

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Section: K-domatic Number Of Special Graphsmentioning

confidence: 90%

The Algorithmic Complexity of k-Domatic Partition of Graphs

Liang

2012

Lecture Notes in Computer Science

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“…The k-domatic number was introduced by Zelinka [11]. Further results on the k-domatic number can be found in the paper [4] by Kämmerling and Volkmann. Let k ≥ 1 be an integer. Following Kämmerling and Volkmann [5], a Roman k-dominating function (briefly RkDF) on a graph G is a labeling f : V (G) → {0, 1, 2} such that every vertex with label 0 has at least k neighbors with label 2.…”

Section: Introductionmentioning

confidence: 99%

The Roman (k,k)-domatic number of a graph

Kazemi¹,

Sheikholeslami

Volkmann

2020

Preprint

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Let k be a positive integer. A Roman k-dominating function on a graph G is a labeling f : V (G) −→ {0, 1, 2} such that every vertex with label 0 has at least k neighbors with label 2. A set {f 1 , f 2 , . . . , f d } of distinct Roman k-dominating functions on G with the property thatThe maximum number of functions in a Roman (k, k)-dominating family on G is the Roman (k, k)-domatic number of G, denoted by d k R (G). Note that the Roman (1, 1)-domatic number d 1 R (G) is the usual Roman domatic number d R (G). In this paper we initiate the study of the Roman (k, k)-domatic number in graphs and we present sharp bounds for d k R (G). In addition, we determine the Roman (k, k)-domatic number of some graphs. Some of our results extend those given by Sheikholeslami and Volkmann in 2010 for the Roman domatic number.

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“…Nesta seção, vamos definir variantes do problema da partição dominante e apresentar alguns dos resultados existentes na literatura. Existem outras variações deste problema sendo estudadas [69,91,93,99].…”

Section: Variantes Da Partição Dominanteunclassified

Um estudo sobre conjuntos cliques-dominantes em grafos

Sousa¹,

Campos²

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A dominating set of a graph G =(V (G),E(G)) is a set S ⊆ V (G) such that every vertex of G belongs to S or is adjacent to at least one vertex in S.T h edominating set problem consists of determining the minimum number of vertices in a dominating set of G.M a n y real-world applications can be modelled as dominating set problems and some of then led to the appearance of variants of the original problem. A dominating clique set is a dominating set that is also a clique. The dominating clique problem consists of determining the minimum number of vertices in a dominating clique set of a graph G.I n 1 9 7 7 , Cockayne and Hedetniemi [41] defined a domatic partition of a graph G =(V (G),E(G)) as a partition of V (G) in dominating sets. The domatic partition problem consists of determining the maximum number of parts in a domatic partition of a graph. A natural extension of this problem consists of considering partitions of V (G) in dominating sets with aditional properties. In particular, a clique domatic partition is a partition of V (G) into dominating clique sets; and the clique domatic partition problem,t h ep r o b l e mo f determining the clique domatic number of G, d cl (G):=max{|P| : P is a clique domatic partition of G},w h e nG has at least one clique domatic partition.In this work, we approach the dominating clique problem and the clique domatic partition problem. For the dominating clique problem we review some existing results in the literature and formulate a conjecture, which establishes a sufficient condition for the existence of a dominating clique set in a graph. On the clique domatic partition problem, we obtain results in some classes of graphs. In particular, we characterize the bipartite graphs and powers of cycles which have clique domatic partitions and for these graphs we determine the clique domatic number. Similar results are also obtained for the class of graphs generated by the cartesian product operation and the class of graphs generated by the direct product of complete graphs. ix x ResumoUm conjunto dominante em um grafo G =( V (G),E(G)) é um conjunto S ⊆ V (G), tal que todo vértice do grafo, ou pertence a S, ou é adjacente a pelo menos um elemento de S.Oproblema do conjunto dominante consiste em determinar a cardinalidade de um conjunto dominante mínimo em um grafo G. Muitas aplicações podem ser modeladas como problemas de conjuntos dominantes e algumas delas levaram ao surgimento de variantes do problema original. O problema da clique-dominante consiste em determinar ac a r d i n a l i d a d ed eu mc o n j u n t oc l i q u e -d o m i n a n t em í n i m oe mu mg r a f oG. Um conjunto clique-dominante é um conjunto dominante que tem a propriedade adicional de ser uma clique. Em 1977, Cockayne e Hedetniemi [41] definiram uma partição dominante de um grafo G =( V (G),E(G)) como uma partição de V (G) em conjuntos dominantes. O problema da partição dominante consiste em determinar a cardinalidade máxima de uma partição dominante. Uma extensão natural deste problema consiste em considerar par...

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The k-domatic number of a graph

Cited by 7 publications

References 6 publications

The Algorithmic Complexity of k-Domatic Partition of Graphs

The Algorithmic Complexity of k-Domatic Partition of Graphs

The Roman (k,k)-domatic number of a graph

Um estudo sobre conjuntos cliques-dominantes em grafos

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