S.V. Divya Rashmi - Department of Mathematics, Vidyavardhaka College of Engineering, Mysuru ۵۷۰۰۰۲, Karnataka, India.
Subramanian Arumugam - National Centre for Advanced Research in Discrete Mathematics, Kalasalingam University, Anand Nagar, Krishnankoil-۶۲۶ ۱۲۶, Tamil Nadu, India.
Kiran R. Bhutani - Department of Mathematics, The Catholic University of America, Washington, D.C. ۲۰۰۶۴, USA.
Peter Gartland - Department of Mathematics, The Catholic University of America, Washington, D.C. ۲۰۰۶۴, USA.

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_۱=(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.

Secure domination, perfect secure domination, secure domination number, perfect secure domination number