Identifying codes in line graphs

Florent Foucaud (1) , Sylvain Gravier (2) , Reza Naserasr (1) , Aline Parreau (2) , Petru Valicov (1)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique
2 IF - Institut Fourier

An identifying code of a graph is a subset of its vertices such that every vertex of the graph is uniquely identified by the set of its neighbours within the code. We study the edge-identifying code problem, i.e. the identifying code problem in line graphs. If $\ID(G)$ denotes the size of a minimum identifying code of an identifiable graph $G$, we show that the usual bound $\ID(G)\ge \lceil\log_2(n+1)\rceil$, where $n$ denotes the order of $G$, can be improved to $\Theta(\sqrt{n})$ in the class of line graphs. Moreover, this bound is tight. We also prove that the upper bound $\ID(\mathcal{L}(G))\leq 2|V(G)|-5$, where $\mathcal{L}(G)$ is the line graph of $G$, holds (with two exceptions). This implies that a conjecture of R.~Klasing, A.~Kosowski, A.~Raspaud and the first author holds for a subclass of line graphs. Finally, we show that the edge-identifying code problem is NP-complete, even for the class of planar bipartite graphs of maximum degree~3 and arbitrarily large girth.

Mots clés: Identifying codes; Dominating sets; Line graphs; NP-completeness

Domaines: Combinatoire [math.CO]; Mathématique discrète [cs.DM]

Florent Foucaud, Sylvain Gravier, Reza Naserasr, Aline Parreau, Petru Valicov. Identifying codes in line graphs. Journal of Graph Theory, 2013, 73 (4), pp.425-448. ⟨10.1002/jgt.21686⟩. ⟨hal-00605406v2⟩