On Minimum Identifying Codes in Some Cartesian Product Graphs

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 γID(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma ^\mathrm{ID}(G)$$\end{document}. In this paper, we give upper and lower bounds for the ID code number of the prism of a graph, or G□K2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G\Box K_2$$\end{document}. In particular, we show that γID(G□K2)≥γID(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma ^\mathrm{ID}(G \Box K_2) \ge \gamma ^\mathrm{ID}(G)$$\end{document} 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 γID(G□K2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma ^\mathrm{ID}(G\Box K_2)$$\end{document}.