Self-identifying codes in direct products of complete graphs with paths and cycles

TL;DR

This paper studies the minimum size of self-identifying codes in direct product graphs of complete graphs with paths and cycles, providing asymptotically tight bounds that depend on the parameters.

Contribution

It derives bounds on the size of self-identifying codes in $K_m\times P_n$ and $K_m\times C_n$, extending understanding of these codes in product graphs.

Findings

Bounds are linear in $n$ with coefficients depending on $m$.

Bounds are asymptotically tight.

Results closely match identifying codes in similar graphs.

Abstract

Identifying codes were introduced by Karpovsky et al. as dominating sets $S\subseteq V(G)$ satisfying $N[u]\cap S \neq N[v]\cap S$ for any distinct vertices $u,v$. Later, Junnila et al. introduced the concept of \emph{self-identifying codes} (previously called $(1,\leq1)^+$-identifying codes in earlier work), a dominating set $S\subseteq V(G)$ such that $\bigcap_{c\in N[u]\cap S} N[c] = \{u\}$ for every vertex $u$. In this paper, we obtain bounds on the minimum size of a self-identifying code in the direct products $K_m\times P_n$ and $K_m\times C_n$ that are linear in $n$ with coefficients depending on $m$, and these bounds are asymptotically tight. In particular, for $K_m\times P_n$ with $m,n\ge3$, our bounds closely approaches the size of an identifying code in the same graph, as determined by Shinde and Waphare.