On Error-detecting Open-locating-dominating sets

TL;DR

This paper investigates error-detecting open-locating-dominating sets in graphs, proving NP-completeness, exploring extremal graphs, and characterizing cubic graphs that admit such sets, advancing understanding of fault-tolerant vertex identification.

Contribution

It introduces and analyzes error-detecting open-locating-dominating sets, providing NP-completeness proof, extremal graph results, and characterizations for cubic graphs.

Findings

NP-completeness of the problem established

Characterization of cubic graphs with error-detecting sets

Identification of extremal graphs for the property

Abstract

An open-dominating set S for a graph G is a subset of vertices where every vertex has a neighbor in S. An open-locating-dominating set S for a graph G is an open-dominating set such that each pair of distinct vertices in G have distinct set of open-neighbors in S. We consider a type of a fault-tolerant open-locating dominating set called error-detecting open-locating-dominating sets. We present more results on the topic including its NP-completeness proof, extremal graphs, and a characterization of cubic graphs that permit an error-detecting open-locating-dominating set.