Edge identifying codes
Résumé
We study the edge identifying code problem, i.e. the identifying code problem in line graphs. If $\M(G)$ denotes the size of a minimum identifying code of a graph $G$, we show that the usual bound $\M(G)\ge \lceil\log_2(n+1)\rceil$, 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 $\M(\mathcal{L}(G))\leq 2\cdot|V(G)|-4$ holds. 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.