Identifying open codes in trees and 4-cycle-free graphs of given maximum degree

Abstract

An identifying open code of a graph $G$ is a set $S$ of vertices that is both a separating open code (that is, $N_G(u)\cap S\ne N_G(v)\cap S$ for all distinct vertices $u$ and $v$ in $G$) and a total dominating set (that is, $N(v)\cap S\ne \varnothing$ for all vertices $v$ in $G$). Such a set exists if and only if the graph $G$ is open twin-free and isolate-free; and the minimum cardinality of an identifying open code in an open twin-free and isolate-free graph $G$, its identifying open code number, is denoted by ${\gamma }^{\mathrm{IOC}}(G)$. We study the identifying open code number of a graph, in relation with its order and its maximum degree. For $\Delta$ a fixed integer at least $3$, we show that if $G$ is a connected graph of order $n\ge 5$ that contains no $4$-cycle and is open twin-free with maximum degree bounded above by $\Delta$, then ${\gamma }^{\mathrm{IOC}}(G)\le \left(\frac{2\Delta -1}{2\Delta }\right)n$, unless $G$ is obtained from a star $K_{1,\Delta }$ by subdividing every edge exactly once. Moreover, we show that the bound is best possible by constructing graphs that reach the bound when $\Delta =3$, and nearly best possible by another construction when $\Delta \ge 4$, with identifying open code numbers $\left(\frac{2\Delta -4}{2\Delta -3}\right)n$.