Algorithm and Hardness Results for Outer-connected Dominating Set in Graphs

Panda, B. S.; Pandey, Arti

doi:10.7155/jgaa.00334

Cited by 13 publications

(4 citation statements)

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“…In particular, there exist graph classes for which the first problem is polynomial-time solvable whereas the second problem is NP-complete and vice versa. Similar study has been made between domination and other domination parameters in [14,20,21]. The DOMINATION problem is linear time solvable for doubly chordal graphs [3], but the SCDM problem is NP-complete for this class of graphs which is proved in section 2.1.…”

Section: Complexity Difference In Domination and Secure Connected Dom...mentioning

confidence: 67%

Algorithmic aspects of secure connected domination in graphs

Kumar¹,

Reddy²

2021

Discuss. Math. Graph Theory

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A dominating set S is an Isolate Dominating Set (IDS ) if the induced subgraph G[S] has at least one isolated vertex. In this paper, we initiate the study of new domination parameter called, isolate secure domination. An isolate dominating set S ⊆ V is an isolate secure dominating set (ISDS ), if for each vertex u ∈ V \S, there exists a neighboring vertex v of u in S such that (S \ {v}) ∪ {u} is an IDS of G. The minimum cardinality of an ISDS of G is called as an isolate secure domination number, and is denoted by γ 0s (G). Given a graph G = (V, E) and a positive integer k, the ISDM problem is to check whether G has an isolate secure dominating set of size at most k. We prove that ISDM is NP-complete even when restricted to bipartite graphs and split graphs. We also show that ISDM can be solved in linear time for graphs of bounded tree-width.

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Section: Complexity Difference In Domination and Secure Connected Dom...mentioning

confidence: 67%

Algorithmic aspects of secure connected domination in graphs

Kumar¹,

Reddy²

2021

Discuss. Math. Graph Theory

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“…A dominating set D is called an outer-connected dominating set if G[V \D] is connected. The concept of outer-connected domination was introduced by Cyman in [6] and further studied in [1,[19][20][21]24]. A set D ⊆ V is a total dominating set of G = (V , E) if D totally dominates V .…”

Section: Introductionmentioning

confidence: 99%

Complexity of total outer-connected domination problem in graphs

Panda

Pandey

2016

Discrete Applied Mathematics

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is to find a total outer-connected dominating set of minimum cardinality. The Total Outer-connected Domination Decision problem, the decision version of the Minimum Total Outer-connected Domination problem, is known to be NP-complete for general graphs. In this paper, we strengthen this NP-completeness result by showing that the Total Outer-connected Domination Decision problem remains NP-complete for chordal graphs, split graphs, and doubly chordal graphs. We prove that the Total Outer-connected Domination Decision problem can be solved in linear time for bounded tree-width graphs. We, then, propose a linear time algorithm for computing the total outer-connected domination number, the cardinality of a minimum total outerconnected dominating set, of a tree. We prove that the Minimum Total Outer-connected Domination problem cannot be approximated within a factor of (1 − ε) ln |V | for any ε > 0, unless NP ⊆ DTIME (|V | O(log log |V |) ). Finally, we show that the Minimum Total Outerconnected Domination problem is APX-complete for bounded degree graphs.

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“…Since overload of the servers is a bottleneck in such a network, every client must be able to communicate to another client directly without interrupting any server. The smallest group of servers with these properties is a minimum outer-connected dominating set for the graph representing the computer network [13]. We make use the following result related to the outer-connected domination number in this paper.…”

Section: Introduction and Preliminarymentioning

confidence: 99%

On the outer-connected domination number for graph products

Hashemipour

Hooshmandasl

Shakiba

2019

Journal of Information and Optimization Sciences

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An outer-connected dominating set for an arbitrary graph G is a setD ⊆ V such thatD is a dominating set and the induced subgraph G[V \D] be connected. In this paper, we focus on the outerconnected domination number of the product of graphs. We investigate the existence of outer-connected dominating set in lexicographic product and Corona of two arbitrary graphs, and we present upper bounds for outer-connected domination number in lexicographic and Cartesian product of graphs. Also, we establish an equivalent form of the Vizing's conjecture for outer-connected domination number in lexicographic and Cartesian product asγc(G • K)γc(H • K) ≤γc(G H) • K. Furthermore, we study the outer-connected domination number of the direct product of finitely many complete graphs.

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Algorithm and Hardness Results for Outer-connected Dominating Set in Graphs

Cited by 13 publications

References 0 publications

Algorithmic aspects of secure connected domination in graphs

Algorithmic aspects of secure connected domination in graphs

Complexity of total outer-connected domination problem in graphs

On the outer-connected domination number for graph products

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