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, $N(v) \cap S \ne \emptyset$ 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 $G$ is denoted by $γ^{\rm {\small IOC}}(G)$. We study the smallest size of an identifying open code of a graph, in relation with its order and its maximum degree. For $Δ$ a fixed integer at least $3$, 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 $Δ$, then we show that $γ^{\rm {\small IOC}}(G) \le \left( \frac{2Δ- 1}Δ \right) n$, unless $G$ is obtained from a star $K_{1,Δ}$ by subdividing every edge exactly once. Moreover, we show that the bound is best possible by constructing graphs that reach the bound.