Identifying Codes in Trees and Planar Graphs

We deal with a few issues related to the problem of finding the minimum size of an identifying code in a graph. First, we provide a linear algorithm which computes an identifying code with minimal size in a given tree. Second, we extend known NP-hardness results by showing that this problem remains NP-hard in the class of planar graphs with arbitrary high girth and maximal degree at most 4. We give similar results for the problem of finding the minimum size of an ( r , ⩽ ℓ ) -identifying code, for all r ⩾ 1 and ℓ ∈ { 1 ; 2 } .