Shahid Beheshti University Categories and General Algebraic Structures with Applications 2345-5853 7 Special Issue on the Occasion of Banaschewski's 90th Birthday (II) 2017 07 01 Perfect secure domination in graphs 125 140 44926 EN S.V. Divya Rashmi Department of Mathematics, Vidyavardhaka College of Engineering, Mysuru 570002, Karnataka, India. Subramanian Arumugam National Centre for Advanced Research in Discrete Mathematics, Kalasalingam University, Anand Nagar, Krishnankoil-626 126, Tamil Nadu, India. Kiran R. Bhutani Department of Mathematics, The Catholic University of America, Washington, D.C. 20064, USA. Peter Gartland Department of Mathematics, The Catholic University of America, Washington, D.C. 20064, USA. Journal Article 2016 10 16 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. https://cgasa.sbu.ac.ir/article_44926_4a0432bd29e2bbab421183f554f06243.pdf