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Algorithmic aspect of ktuple domination in graphs
 Taiwanese J. Math
"... Abstract. In a graph G, a vertex is said to dominate itself and all of its neighbors. For a fixed positive integer k, the ktuple domination problem is to find a minimum sized vertex subset such that every vertex in the graph is dominated by at least k vertices in this set. The present paper studies ..."
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Abstract. In a graph G, a vertex is said to dominate itself and all of its neighbors. For a fixed positive integer k, the ktuple domination problem is to find a minimum sized vertex subset such that every vertex in the graph is dominated by at least k vertices in this set. The present paper studies the ktuple domination problem in graphs from an algorithmic point of view. In particular, we give a lineartime algorithm for the 2tuple domination problem in trees by employing a labeling method. 1.
The Weighted Independent Domination Problem Is NPComplete for Chordal Graphs
 Discrete Applied Mathematics
"... The concept of domination in graph theory is a natural model for many location problems in operations research. In a graph G = (V, E), a dominating set is a subset D of V such that every vertex not in D is adjacent to at least one vertex in D. The domination problem is to determine, for a graph G an ..."
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The concept of domination in graph theory is a natural model for many location problems in operations research. In a graph G = (V, E), a dominating set is a subset D of V such that every vertex not in D is adjacent to at least one vertex in D. The domination problem is to determine, for a graph G and a positive integer k, if G has a dominating set of size at most k. Domination and its variations have been extensively studied in the literature (see [1, 4, 5]). Domination and most of its variations are NPcomplete for chordal graphs (even for the subclass of split graphs) with the exception of independence domination (see [3]). On the other hand, an unpublished proof for the NPcompleteness of the weighted independent domination in chordal graphs by the author twenty years ago (see [3, 2]) has been queried from time to time. Someone even claimed that he has an efficient algorithm for the problem. The purpose of this paper is to make a record of the proof for those who are interested in. Recall that an independent dominating set is a dominating set whose elements are pairwise nonadjacent. For the weighted version, each vertex is associated with a
Nonsplit and Inverse Nonsplit Domination Numbers in the Join and Corona of Graphs
"... A dominating set D of a graph G = (V, E) is nonsplit dominating set if 〈V \ D 〉 is connected. The nonsplit domination number of G is the minimum cardinality of a nonsplit dominating set in G. Let D be a minimum dominating set in G. If a subset D ′ of V \ D is dominating in G, then D ′ is called a ..."
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A dominating set D of a graph G = (V, E) is nonsplit dominating set if 〈V \ D 〉 is connected. The nonsplit domination number of G is the minimum cardinality of a nonsplit dominating set in G. Let D be a minimum dominating set in G. If a subset D ′ of V \ D is dominating in G, then D ′ is called an inverse 〈 dominating set with respect to D. Furthermore, if V \ D is connected, then D is called an inverse nonsplit dominating set. The inverse nonsplit domination number of G is the minimum cardinality of an inverse nonsplit dominating set in G. In this paper, characterization of nonsplit dominating sets in the join and corona of two graphs are presented. Furthermore, explicit formulas for determining the nonsplit and inverse nonsplit domination numbers of these graphs are also determined.
Perfect Domination with Forbidden Vertices on lkstarlike Graphs and Trees
"... Let G(V, E) be a graph with nvertexset V and medgeset E. Meanwhile, each vertex v is associated with a nonnegative weight W(v). Given a subset F of V, this paper studies the problem of finding a perfect dominating set D such that δ(D) = ∑ W ( D) is v∈D minimized under the restriction that D ⊆ ..."
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Let G(V, E) be a graph with nvertexset V and medgeset E. Meanwhile, each vertex v is associated with a nonnegative weight W(v). Given a subset F of V, this paper studies the problem of finding a perfect dominating set D such that δ(D) = ∑ W ( D) is v∈D minimized under the restriction that D ⊆ (V F). The vertices in F are called forbidden vertices. We first define a new class of graphs, called lkstarlike graphs, k ≥ 1, which is a super class of the class of starlike graphs. Then, we show that the problem is NPHard on lkstarlike graphs, k ≥ 2, but lineartime solvable when k = 1. Finally, an O(n)time algorithm for the problem on trees is designed.
The 29th Workshop on Combinatorial Mathematics and Computation Theory The mixed dominating set problem is MAX SNPhard
"... Given a graph G =(V,E), a mixed dominating set MD of G is defined to be a subset of V ∪ E such that every element in {(V ∪E)\MD} is either adjacent or incident to an element of MD. The mixed dominating set problem is to find a mixed dominating set with minimum cardinality. This problem is NPhard. I ..."
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Given a graph G =(V,E), a mixed dominating set MD of G is defined to be a subset of V ∪ E such that every element in {(V ∪E)\MD} is either adjacent or incident to an element of MD. The mixed dominating set problem is to find a mixed dominating set with minimum cardinality. This problem is NPhard. In this paper, we prove that this problem is MAX SNPhard.