ON GENERALIZED DISTANCE SECURE SETS IN GRAPHS
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Let S be a non empty subset of vertex set. For any v ∈ S, the vertex v and its neighbors inside S are called defenders of v, whereas those lying outside S are attackers of v with respect to S. An attack on S is a collection of mutually disjoint sets of attackers of vertices in S, whereas a defense is that of defenders. An attack on S is defendable if there is a defense of S such that for every vertex in S, the cardinality of corresponding defending set is more than or equal to that of corresponding attacking set. The set S is a secure set if every attack on S is defendable. An ultra secure set is a secure set in which every attack is defendable by a single defense of S. In this paper, various types of generalized distance secure sets are introduced by considering attackers and defenders from distances more than 1. A characterization of k-distance ultra secure sets is obtained and some properties of generalized distance secure sets are discussed.
Abstract
Let S be a non empty subset of vertex set. For any v ∈ S, the vertex v and its neighbors inside S are called defenders of v, whereas those lying outside S are attackers of v with respect to S. An attack on S is a collection of mutually disjoint sets of attackers of vertices in S, whereas a defense is that of defenders. An attack on S is defendable if there is a defense of S such that for every vertex in S, the cardinality of corresponding defending set is more than or equal to that of corresponding attacking set. The set S is a secure set if every attack on S is defendable. An ultra secure set is a secure set in which every attack is defendable by a single defense of S. In this paper, various types of generalized distance secure sets are introduced by considering attackers and defenders from distances more than 1. A characterization of k-distance ultra secure sets is obtained and some properties of generalized distance secure sets are discussed.
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