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Identifying Codes in Line Graphs


  • Contract grant sponsor: ANR Project IDEA; Contract grant number: ANR-08-EMER-007.


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 neighbors within the code. We study the edge-identifying code problem, i.e. the identifying code problem in line graphs. If math formula denotes the size of a minimum identifying code of an identifiable graph G, we show that the usual bound math formula, where n denotes the order of G, can be improved to math formula in the class of line graphs. Moreover, this bound is tight. We also prove that the upper bound math formula, where math formula 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.