Minimal identifying codes in trees and planar graphs with large girth

Abstract

AbstractLet G be a finite undirected graph with vertex set V(G). If v∈V(G), let N[v] denote the closed neighbourhood of v, i.e. v itself and all its adjacent vertices in G. An identifying code in G is a subset C of V(G) such that the sets N[v]∩C are nonempty and pairwise distinct for each vertex v∈V(G). We consider the problem of finding the minimum size of an identifying code in a given graph, which is known to be NP-hard. We give a linear algorithm that solves it in the class of trees, but show that the problem remains NP-hard in the class of planar graphs with arbitrarily large girth