The complexity of monitoring a network with both watchers and listeners

Abstract

We consider the problem of detecting an intruder in a network where there are two types of detecting devices available. One device can determine the distance from itself to the intruder and the other can see the vertex it occupies as well as all adjacent vertices. The problem is to determine how many devices are required and where they should be placed in order to determine a single intruder’s location. We show that on the one hand, finding the minimum number of devices required to do this is easy when the network is a tree with at most one leaf adjacent to any vertex, while on the other hand finding this number is an NP-complete problem even for trees with at most two leaves adjacent to any vertex.