Opuscula Math. 24, no. 2 (2004), 231-234

Opuscula Mathematica

Domination parameters of a graph with added vertex

Maciej Zwierzchowski

Abstract. Let \(G=(V,E)\) be a graph. A subset \(D\subseteq V\) is a total dominating set of \(G\) if for every vertex \(y\in V\) there is a vertex \(x\in D\) with \(xy\in E\). A subset \(D\subseteq V\) is a strong dominating set of \(G\) if for every vertex \(y\in V-D\) there is a vertex \(x\in D\) with \(xy\in E\) and \(\deg _{G}(x)\geq\deg _{G}(y)\). The total domination number \(\gamma _{t}(G)\) (the strong domination number \(\gamma_{S}(G)\)) is defined as the minimum cardinality of a total dominating set (a strong dominating set) of \(G\). The concept of total domination was first defined by Cockayne, Dawes and Hedetniemi in 1980 [Cockayne E. J., Dawes R. M., Hedetniemi S. T.: Total domination in graphs. Networks 10 (1980), 211–219], while the strong domination was introduced by Sampathkumar and Pushpa Latha in 1996 [Pushpa Latha L., Sampathkumar E.: Strong weak domination and domination balance in a graph. Discrete Mathematics 161 (1996), 235–242]. By a subdivision of an edge \(uv\in E\) we mean removing edge \(uv\), adding a new vertex \(x\), and adding edges \(ux\) and \(vx\). A graph obtained from \(G\) by subdivision an edge \(uv\in E\) is denoted by \(G\oplus u_{x}v_{x}\). The behaviour of the total domination number and the strong domination number of a graph \(G\oplus u_{x}v_{x}\) is developed.

Keywords: the total domination number, the strong domination number, subdivision.

Mathematics Subject Classification: 05C69.

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  • Maciej Zwierzchowski
  • Technical University of Szczecin, Institute of Mathematics, al. Piastów 48/49, 70-310 Szczecin, Poland
  • Received: 2003-11-16.
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Maciej Zwierzchowski, Domination parameters of a graph with added vertex, Opuscula Math. 24, no. 2 (2004), 231-234

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