Extremal cardinalities for identifying and locating-dominating codes in graphs

Abstract

Consider a connected undirected graph G = ( V , E ) , a subset of vertices C ⊆ V , and an integer r ⩾ 1 ; for any vertex v ∈ V , let B r ( v ) denote the ball of radius r centred at v , i.e., the set of all vertices linked to v by a path of at most r edges. If for all vertices v ∈ V (respectively, v ∈ V ⧹ C ), the sets B r ( v ) ∩ C are all nonempty and different, then we call C an r-identifying code (respectively, an r-locating-dominating code). We study the extremal values of the cardinality of a minimum r-identifying or r-locating-dominating code in any connected undirected graph G having a given number, n, of vertices. It is known that a minimum r-identifying code contains at least ⌈ log 2 ( n + 1 ) ⌉ vertices; we establish in particular that such a code contains at most n - 1 vertices, and we prove that these two bounds are reached. The same type of results are given for locating-dominating codes.