Open locating-dominating sets in circulant graphs

Abstract

Location detection problems have been studied for a variety of applications including finding faults in multiprocessors, contaminants in public utilities, intruders in buildings and facilities, and for environmental monitoring using wireless sensor networks. In each of these applications, the system or structure can be modeled as a graph, and sensors placed strategically at a subset of vertices can locate and detect anomalies in the system. An open locating-dominating set (OLD-set) is a subset of vertices in a graph in which every vertex in the graph has a non-empty and unique set of neighbors in the subset. Sensors placed at OLD-set vertices can uniquely detect and locate disturbances in a system. These sensors can be expensive and, as a result, minimizing the size of the OLD-set is critical. Circulant graphs, a group of regular cyclic graphs, are often used to model parallel networks. We prove the optimal OLD-set size for a particular circulant graph using Hall's Theorem. We also consider the mixed-weight OLD-set introduced in [R.

Extremal digraphs for open neighbourhood location-domination and identifying codes

“…Since their introduction over a decade ago, OLD sets have been extensively studied, see [10,15,16,19,21,24,26,31] for some papers on the topic. OLD sets are related to locating-dominating sets, defined by Slater in the 1980s [32,33], where the open neighbourhood domination condition is replaced by closed neighbourhood domination, and the location condition is only required for pairs of vertices that are not in the solution set.…”

Characterizing extremal graphs for open neighbourhood location-domination

“…The notion of open neighbourhood location-domination was introduced about a decade ago by Seo and Slater in [19], and various aspects of the problem were subsequently studied in [3,7,8,10,12,15,17,20]. Open neighbourhood locating-dominating sets are related to the concept of identifying codes [14] (where open neighbourhoods are replaced by closed neighbourhoods) and the earlier concept of locating-dominating sets [21,22] (where only vertices not in the solution set need to be uniquely identified).…”