Perfect secure domination in graphs

Document Type: Research Paper

Authors

1 Department of Mathematics, Vidyavardhaka College of Engineering, Mysuru 570002, Karnataka, India.

2 National Centre for Advanced Research in Discrete Mathematics, Kalasalingam University, Anand Nagar, Krishnankoil-626 126, Tamil Nadu, India.

3 Department of Mathematics, The Catholic University of America, Washington, D.C. 20064, USA.

Abstract

Let $G=(V,E)$ be a graph. A subset $S$ of $V$ is a dominating set of $G$ if every vertex in $V\setminus  S$ is adjacent to a vertex in $S.$ A dominating set $S$ is called a secure dominating set if for each $v\in V\setminus S$ there exists $u\in S$ such that $v$ is adjacent to $u$ and $S_1=(S\setminus\{u\})\cup \{v\}$ is a dominating set. If further the vertex $u\in S$ is unique, then $S$ is called a perfect secure dominating set. The minimum cardinality of a perfect secure dominating set of $G$ is called the perfect  secure domination number of $G$ and is denoted by $\gamma_{ps}(G).$ In this paper we initiate a study of this parameter and present several basic results.

Highlights

Dedicated to Bernhard Banaschewski on the occasion of his 90th birthday

Keywords


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