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Keywords:

  • graph theory;
  • identifying codes;
  • planar graphs;
  • complexity;
  • NP-completeness;
  • NP-hardness

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 vV(G) and rgeqslant R: gt-or-equal, slanted1, let us denote by Ir��(v) the set of codewords c∈�� such that d(vc)leqslant R: less-than-or-eq, slantr, where the distance d(vc) is defined as the length of a shortest path between v and c. More generally, for AV(G), we define inline image, 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 (rleqslant R: less-than-or-eq, slantl)-identifying code if one has Ir��(A)≠Ir��(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 (rleqslant R: less-than-or-eq, slantl)-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 rgeqslant R: gt-or-equal, slanted1. We show that it is also NP-hard in the class of planar graphs with maximum degree at most three for all (rl) with rgeqslant R: gt-or-equal, slanted1 and l∈{1, 2}. This shows, in particular, that the problem of computing the minimum size of an (rleqslant R: less-than-or-eq, slant2)-identifying code in a given graph is NP-hard.