Edge identifying codes

Abstract

Abstract We study the edge identifying code problem, i.e. the identifying code problem in line graphs. If γ ID ( G ) denotes the size of a minimum identifying code of a graph G , we show that the usual bound γ ID ( G ) ⩾ ⌈ log 2 ( n + 1 ) ⌉ , where n denotes the order of G , can be improved to Θ ( n ) in the class of line graphs. Moreover this bound is tight. We also prove that the upper bound γ ID ( L ( G ) ) ⩽ 2 ⋅ | 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.