An integer program for Open Locating Dominating sets and its results on the hexagon-triangle infinite grid and other graphs

Abstract

This paper presents an integer linear program (ILP) for the identification of Open Locating Dominating Sets (OLD) of minimum cardinality and presents several results of the ILP on various graphs. The OLD is similar to an identifying code, but for an open neighborhood instead of closed. The OLD was introduced by Peter Slater and Suk J. Seo in 2010 as a method by which one could identify the location of an event at a node where a node in the set can detect events at adjacent nodes, but cannot detect an event at itself. This is perhaps more clear as a series of factories such that those with intrusion detection devices form an identifying code, but an intruder will disable the system at the factory into which she breaks, if so equipped. There are also applications in other areas such as router networks. This paper continues work by Alison Oldham at the College of William and Mary on the development and implementation of an ILP to identify such OLDs on finite graphs. We demonstrate and compare the results on 100 node randomly generated graphs of various constructions; Erdös-Renyi, Geometric and Scale-Free. We find that most graphs have a density near 1/3. We also explore the use of the ILP to generate OLDs for infinite grids, looking specifically at the hexagonal-triangular grid where we discover a new upper bound of 5/12 on the minimum density OLD for this grid. Finally, we extend this ILP to identify locating dominating sets that simultaneously satisfy open and closed neighborhood criteria, or Closed-Open Locating Dominating sets (COLD).