On Minimum Identifying Codes in Some Cartesian Product Graphs

Abstract

An identifying code in a graph is a dominating set that also has the property that the closed neighborhood of each vertex in the graph has a distinct intersection with the set. The minimum cardinality of an identifying code, or ID code, in a graph G is called the ID code number of G and is denoted $$\gamma ^\mathrm{ID}(G)$$ . In this paper, we give upper and lower bounds for the ID code number of the prism of a graph, or $$G\Box K_2$$ . In particular, we show that $$\gamma ^\mathrm{ID}(G \Box K_2) \ge \gamma ^\mathrm{ID}(G)$$ and we show that this bound is sharp. We also give upper and lower bounds for the ID code number of grid graphs and a general upper bound for $$\gamma ^\mathrm{ID}(G\Box K_2)$$ .