Lexicographic identifying codes

An identifying code in a graph is a set of vertices which intersects all the symmetric differences between pairs of neighbourhoods of vertices. Not all graphs have identifying codes; those that do are referred to as twin-free. In this paper, we design an algorithm that finds an identifying code in a twin-free graph on n vertices in O(n) binary operations, and returns a failure if the graph is not twin-free. We also determine an alternative for sparse graphs with a running time of O(nd log n) binary operations, where d is the maximum degree. We also prove that these algorithms can return any identifying code with minimum cardinality, provided the vertices are correctly sorted.