On minimum identifying codes in some Cartesian product graphs

TL;DR

This paper investigates the minimum size of identifying codes in Cartesian product graphs, providing bounds and sharpness results for prisms and grid graphs, advancing understanding of graph identification properties.

Contribution

It establishes new bounds for the ID code number in Cartesian product graphs, especially for prisms and grid graphs, including sharpness of these bounds.

Findings

; ; ; bounds for ; in Cartesian product graphs.

; the bound ; is sharp.

; upper bounds for ; in grid 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 $\gid(G)$. 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 $\gid(G \Box K_2) \ge \gid(G)$ 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 $\gid(G\Box K_2)$.