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

TL;DR

This paper investigates the minimum size of identifying open codes in certain graphs, establishing bounds based on maximum degree and graph structure, and demonstrating the bounds are tight with specific constructions.

Contribution

It provides a new upper bound on the size of identifying open codes in open twin-free, 4-cycle-free graphs with bounded maximum degree, and shows this bound is optimal.

Findings

Bound on identifying open code size: 8rac{2\Delta - 1}{\Delta} n

Graphs reaching the bound are explicitly constructed

Results apply to graphs with maximum degree at least 3, no 4-cycle, and specific structural conditions.

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 $\gamma^{{\rm {\small IOC}}}(G)$. We study the smallest size of an identifying open code of a graph, in relation with its order and its maximum degree. For $\Delta$ 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 $\Delta$, then we show that $\gamma^{{\rm {\small IOC}}}(G) \le \left( \frac{2\Delta - 1}{\Delta} \right) n$, 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.