Locating Mobile Intruders using Dominating Sets

J. Louis Sewell1, Peter J. Slater2
1Mathematical Sciences Department,University of Alabama in Huntsville Huntsville, AL 35899
2Computer Science Department University Of Alabama In Huntsville Huntsville, Al 35899 USA

Abstract

We consider the placement of detection devices at the vertices of a graph \( G \), where a detection device at vertex \( v \) has three possible outputs: there is an intruder at \( v \); there is an intruder at one of the vertices in the open neighborhood \( N(v) \), the set of vertices adjacent to \( v \), but which vertex in \( N(v) \) cannot be determined; or there is no intruder in \( N[v] = N(v) \cup \{v\} \). We introduce the \( 1 \)-step locating-dominating problem of placing the minimum possible number of such detection devices in \( V(G) \) so that the presence of an intruder in \( V(G) \) can be detected, and the exact location of the intruder can be identified, either immediately or when the intruder has moved to an adjacent vertex. Some related problems are introduced.