Monitoring the edges of a graph using distances
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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 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≤ dem(G)≤ n-1 with dem(G)=1 if and only if G is a tree, and dem(G)=n-1 if and only if it is a complete graph. We compute the exact value of dem for grids, hypercubes, and complete bipartite graphs. Then, we relate dem to other standard graph parameters. We show that demG) is lower-bounded by the arboricity of the graph, and upper-bounded by its vertex cover number. It is also upper-bounded by twice its feedback edge set number. Moreover, we characterize connected graphs G with dem(G)=2. Then, we show that determining 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 unlikey to be fixed parameter tractable when parameterized by the solution size.
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