Generalization of the total outer-connected domination in graphs

Rad, Nader Jafari; Volkmann, Lutz

doi:10.1051/ro/2015016

Rad, Nader Jafari ¹ ; Volkmann, Lutz ²

¹ Department of Mathematics, Shahrood University of Technology, P.O. Box 3619995161, Shahrood, Iran.
² Lehrstuhl II für Mathematik, RWTH Aachen University, Templergraben 55, 52056 Aachen, Germany.

RAIRO - Operations Research - Recherche Opérationnelle, Special issue: Research on Optimization and Graph Theory dedicated to COSI 2013 / Special issue: Recent Advances in Operations Research in Computational Biology, Bioinformatics and Medicine, Tome 50 (2016) no. 2, pp. 233-239.

Résumé

Let $G = (V, E)$ be a graph without an isolated vertex. A set $S \subseteq V$ is a total dominating set if $S$ is a dominating set, and the induced subgraph $G [S]$ does not contain an isolated vertex. The total domination number of $G$ is the minimum cardinality of a total dominating set of $G$ . A set $D \subseteq V$ is a total outer-connected dominating set if $D$ is a total dominating set, and the induced subgraph $G [V - D]$ is connected. The total outer-connected domination number of $G$ is the minimum cardinality of a total outer-connected dominating set of $G$ . In this paper we generalize the total outer-connected domination number in graphs. Let $k \geq 1$ be an integer. A set $D \subseteq V$ is a total outer- $k$ -connected component dominating set if $D$ is a total dominating and the induced subgraph $G [V - D]$ has exactly $k$ connected component(s). The total outer- $k$ -connected component domination number of $G$ , denoted by $γ_{t c}^{k} (G)$ , is the minimum cardinality of a total outer- $k$ -connected component dominating set of $G$ . We obtain several general results and bounds for $γ_{t c}^{k} (G)$ , and we determine exact values of $γ_{t c}^{k} (G)$ for some special classes of graphs $G$ .

MR Zbl

DOI : 10.1051/ro/2015016

Classification : 05C69
Mots-clés : Total domination, total outer-connected domination

Affiliations des auteurs :

Rad, Nader Jafari ¹ ; Volkmann, Lutz ²

¹ Department of Mathematics, Shahrood University of Technology, P.O. Box 3619995161, Shahrood, Iran.
² Lehrstuhl II für Mathematik, RWTH Aachen University, Templergraben 55, 52056 Aachen, Germany.

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     title = {Generalization of the total outer-connected domination in graphs},
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     pages = {233--239},
     publisher = {EDP-Sciences},
     volume = {50},
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Rad, Nader Jafari; Volkmann, Lutz. Generalization of the total outer-connected domination in graphs. RAIRO - Operations Research - Recherche Opérationnelle, Special issue: Research on Optimization and Graph Theory dedicated to COSI 2013 / Special issue: Recent Advances in Operations Research in Computational Biology, Bioinformatics and Medicine, Tome 50 (2016) no. 2, pp. 233-239. doi : 10.1051/ro/2015016. http://archive.numdam.org/articles/10.1051/ro/2015016/

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