Density of identifying codes of hexagonal grids with finite number of rows

Abstract

In a graph $G$, a set $C\subseteq V(G)$ is an identifying code if, for all vertices $v$ in $G$, the sets $N[v]\cap C$ are all nonempty and pairwise distinct, where $N[v]$ denotes the closed neighbourhood of $v$. We focus on the minimum density of identifying codes of infinite hexagonal grids $H_k$ with $k$ rows, denoted by $d^*(H_k)$, and present optimal solutions for $k\leq 5$. Using discharging method, we also prove a lower bound in terms of maximum degree for the minimum-density identifying codes of well-behaved infinite graphs. We prove that $d^*(H_2)=9/20$, $d^*(H_3)=6/13\approx 0.4615$, $d^*(H_4)=7/16=0.4375$ and $d^*(H_5)=11/25=0:44$. We also prove that $H_2$ has a unique periodic identifying code with minimum density.