K-Distance Facility Set of a Graph

Abstract

 A set  is called a  -distance facility set of   if any geodesic  joining  and  of length at least , contains some  as an internal vertex of  or  where  and . The minimum cardinality of a -distance facility set of  is called minimum -distance facility set of  (in short -set) and the minimum cardinality is called the  -distance facility number of  is denoted by . In this paper we found relation between  and other graph theoretic parameters, also we found the -distance facility number of paths, cycles and trees.