On the total outer-connected domination in graphs

Favaron, Odile; Karami, Hossein; Sheikholeslami, Seyed Mahmoud

doi:10.1007/s10878-012-9531-6

Cited by 3 publications

(3 citation statements)

References 4 publications

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“…A TDS X is said to be a total outer-connected dominating set if the subgraph induced by V(G) \ X is connected. The total outer-connected domination number of G, denoted by γ toc (G), is the minimum cardinality among all total outer-connected dominating sets of G. This parameter was introduced by Cyman in [18] and studied further in [19][20][21].…”

Section: Some Additional Concepts and Notationmentioning

confidence: 99%

On the Secure Total Domination Number of Graphs

2019

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A total dominating set D of a graph G is said to be a secure total dominating set if for every vertex u ∈ V ( G ) \ D , there exists a vertex v ∈ D , which is adjacent to u, such that ( D \ { v } ) ∪ { u } is a total dominating set as well. The secure total domination number of G is the minimum cardinality among all secure total dominating sets of G. In this article, we obtain new relationships between the secure total domination number and other graph parameters: namely the independence number, the matching number and other domination parameters. Some of our results are tight bounds that improve some well-known results.

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Section: Some Additional Concepts and Notationmentioning

confidence: 99%

On the Secure Total Domination Number of Graphs

2019

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“…The total outer-connected domination number of a graph G, denoted by γ toc (G), is the cardinality of a minimum total outer-connected dominating set of G. The concept of total outer-connected domination was introduced by Cyman in [7] and further studied in [8,10,13,18]. This problem has applications in computer networks.…”

Section: Introductionmentioning

confidence: 99%

“…The Minimum Total Outer-connected Domination (MTOCD) problem is to find a total outer-connected dominating set of minimum cardinality of the input graph G. Given a positive integer k and a graph G = (V , E), the Total Outer-connected Domination Decision (TOCDD) problem is to decide whether G has a total outer-connected dominating set of cardinality at most k. The Total Outer-connected Domination Decision problem is known to be NP-complete for general graphs and even for bipartite graphs [10]. In this paper, we strengthen the NP-completeness result of the TOCDD problem by showing that this problem remains NP-complete for chordal graphs, split graphs, and doubly chordal graphs.…”

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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