**International Transactions in Operational Research**

# Complexity results for identifying codes in planar graphs

## Abstract

Let *G* be a simple, undirected, connected graph with vertex set *V*(*G*) and ��⊆*V*(*G*) be a set of vertices whose elements are called *codewords*. For *v*∈*V*(*G*) and *r*1, let us denote by *I _{r}*

^{��}(

*v*) the set of codewords

*c*∈�� such that

*d*(

*v*,

*c*)

*r*, where the distance

*d*(

*v*,

*c*) is defined as the length of a shortest path between

*v*and

*c*. More generally, for

*A*⊆

*V*(

*G*), we define , which is the set of codewords whose minimum distance to an element of

*A*is at most

*r*. If

*r*and

*l*are positive integers,

*��*is said to be an (

*r*,

*l*)-identifying code if one has

*I*

_{r}

^{��}(

*A*)≠

*I*

_{r}^{��}(

*A*′) whenever

*A*and

*A*′ are distinct subsets of

*V*(

*G*) with at most

*l*elements. We consider the problem of finding the minimum size of an (

*r*,

*l*)-identifying code in a given graph. It is already known that this problem is NP-hard in the class of all graphs when

*l*=1 and

*r*1. We show that it is also NP-hard in the class of planar graphs with maximum degree at most three for all (

*r*,

*l*) with

*r*1 and

*l*∈{1, 2}. This shows, in particular, that the problem of computing the minimum size of an (

*r*, 2)-identifying code in a given graph is NP-hard.