Monitoring the Edges of a Graph Using Distances

Abstract

We introduce a new graph-theoretic concept in the area of network monitoring. A set M of vertices of a graph G is a distance-edge-monitoring set if for every edge e of G, there is a vertex x of M and a vertex y of G such that e belongs to all shortest paths between x and y. We denote by $\mathrm {dem}(G)$ the smallest size of such a set in G. The vertices of M represent distance probes in a network modeled by G; when the edge e fails, the distance from x to y increases, and thus we are able to detect the failure. It turns out that not only we can detect it, but we can even correctly locate the failing edge.

In this paper, we initiate the study of this new concept. We show that for a nontrivial connected graph G of order n, $1\le \mathrm {dem}(G)\le n-1$ with $\mathrm {dem}(G)=1$ if and only if G is a tree, and $\mathrm {dem}(G)=n-1$ if and only if it is a complete graph. We compute the exact value of $\mathrm {dem}$ for grids, hypercubes, and complete bipartite graphs.

Then, we relate $\mathrm {dem}$ to other standard graph parameters. We show that $\mathrm {dem}(G)$ is lower-bounded by the arboricity of the graph, and upper-bounded by its vertex cover number. It is also upper-bounded by five times its feedback edge set number.

Then, we show that determining $\mathrm {dem}(G)$ for an input graph G is an NP-complete problem, even for apex graphs. There exists a polynomial-time logarithmic-factor approximation algorithm, however it is NP-hard to compute an asymptotically better approximation, even for bipartite graphs of small diameter and for bipartite subcubic graphs. For such instances, the problem is also unlikely to be fixed parameter tractable when parameterized by the solution size.

This paper is dedicated to the memory of Mirka Miller, who sadly passed away in January 2016. This research was started when Mirka Miller and Joe Ryan visited Ralf Klasing at the LaBRI, University of Bordeaux, in April 2015.