LOCATING-DOMINATING SETS ON ARBITRARY GRAPHS Maya Satratzemi Department of Applied Informatics, University of Macedonia 156 Egnatia Str., P.O.Box 1591, 54006, Thessalonki, Greece Email: maya@uom.gr ABSTRACT A subset S of the vertex set V of a graph G is a Dominating Set if for every S V v − ∈ there exists a vertex S u ∈ such that u is adjacent to v . The neighborhood () v N of a vertex V v ∈ is the set of all vertices adjacent to v . A dominating set is called a Locating- Dominating set if for every two vertices S V w v − ∈ , it holds that: ( ) ( ) S w N S v N ∩ ≠ ∩ . The cardinality of the Minimum Locating – Dominating set is called the location – domination number and is denoted by ( ) G L γ . In this paper we develop a tree search algorithm, which finds the location – domination number on arbitrary graphs. Also we develop a tree search algorithm, which determines the location – domination number on arbitrary graphs. The properties concerning the notion of Minimum dominating and locating set are used in the branching and backtracking process. Finally, a computational experiment on random graphs of different sizes is presented. Keywords: Locating-Dominating set, tree search algorithm, branch and bound 1. INTRODUCTION A subset S of the vertex set V of a graph G [3] is a Dominating Set if for every S V v − ∈ there exists a vertex S u ∈ such that u is adjacent to v [4, 5]. The neighborhood () v N of a vertex V v ∈ is the set of all vertices adjacent to v . A Dominating Set is called a Locating-Dominating set if for every two vertices S V w v − ∈ , it holds that: () ( ) S w N S v N ∩ ≠ ∩ . Thus, with a Locating - Dominating set S every vertex in S V − is dominated by a distinct subset of the vertices of S . The cardinality of the Minimum Locating – Dominating Set is called the location – domination number and is denoted by ( ) G L γ . In [7], Slater first introduces the concept of locating – dominating sets as well as expressing an algorithm in order to find the minimum cardinality for Locating – Dominating Set in an acyclic graph. He came up with the idea of combining the concepts of dominating with locating in an attempt to analyze safeguards systems, such as a study on fire protection in a nuclear power plant. In any such application, a vertex can represent a room, hallway and so on. Any two areas physically adjacent to each other, or even within sight or sound of one other, can be connected at each edge. Keeping in mind that the main function of safeguards systems is to be able to “detect” some object (a fire, a saboteur, an intruder) a collection of vertices must be determined which will enable the detection devices to be positioned. In this way, if there is an object at any vertex of the graph, it can be detected and its position specifically identified. To be able to detect an object at any vertex in V(G), a