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.

Full text (pdf)

  • Maciej Zwierzchowski
  • Technical University of Szczecin, Institute of Mathematics, al. Piastów 48/49, 70-310 Szczecin, Poland
  • Received: 2003-11-16.
Opuscula Mathematica - cover

Cite this article as:
Maciej Zwierzchowski, Domination parameters of a graph with added vertex, Opuscula Math. 24, no. 2 (2004), 231-234

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.