Graphs Admitting (1, ≤ 2)-identifying Codes

A (1,≤ 2)-identifying code is a subset of the vertex set C of a graph such that each pair of vertices intersects C in a distinct way. This has useful applications in locating errors in multiprocessor networks and threat monitoring. At the time of writing, there is no simply-stated rule that will indicate if a graph is (1,≤ 2)-identifiable. As such, we discuss properties that must be satisfied by a valid (1,≤ 2)-identifying code, characteristics of a graph which preclude the existence of a (1,≤ 2)-identifying code, and relationships between the maximum degree and order of (1,≤ 2)-identifiable graphs. Additionally, we show that (1,≤ 2)-identifiable graphs have no forbidden induced subgraphs and provide a list of (1,≤ 2)-identifiable graphs with minimum (1,≤ 2)-identifying codes indicated.