Secure resolving sets in a graph

Abstract

Let G = (V, E) be a simple, finite, and connected graph. A subset S = (u1, u2, . . ., uk) of V(G) is called a resolving set (locating set) if for any x ∈ V(G), the code of x with respect to S that is denoted by CS (x), which is defined as CS (x) = (d(u1, x), d(u2, x), .., d(uk, x)), is different for different x. The minimum cardinality of a resolving set is called the dimension of G and is denoted by dim(G). A security concept was introduced in domination. A subset D of V(G) is called a dominating set of G if for any v in V - D, there exists u in D such that u and v are adjacent. A dominating set D is secure if for any u in V - D, there exists v in D such that (D - (v)) ∪ (u) is a dominating set. A resolving set R is secure if for any s ∈ V - R, there exists r ∈ R such that (R - (r)) ∪ (s) is a resolving set. The secure resolving domination number is defined, and its value is found for several classes of graphs. The characterization of graphs with specific secure resolving domination number is also done.