Monitoring the edges of a graph using distances

نویسندگان

چکیده

We introduce a new graph-theoretic concept in the area of network monitoring. A set $M$ vertices graph $G$ is \emph{distance-edge-monitoring set} if for every edge $e$ $G$, there vertex $x$ and $y$ such that belongs to all shortest paths between $y$. denote by $dem(G)$ smallest size $G$. The represent distance probes modeled $G$; when fails, from increases, thus we are able detect failure. It turns out not only can it, but even correctly locate failing edge. In this paper, initiate study concept. show nontrivial connected order $n$, $1\leq dem(G)\leq n-1$ with $dem(G)=1$ tree, $dem(G)=n-1$ it complete graph. compute exact value $dem$ grids, hypercubes, bipartite graphs. Then, relate other standard parameters. $demG)$ lower-bounded arboricity graph, upper-bounded its cover number. also twice feedback Moreover, characterize graphs $dem(G)=2$. determining an input NP-complete problem, apex There exists polynomial-time logarithmic-factor approximation algorithm, however NP-hard asymptotically better approximation, small diameter subcubic For instances, problem unlikey be fixed parameter tractable parameterized solution size.

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ژورنال

عنوان ژورنال: Discrete Applied Mathematics

سال: 2022

ISSN: ['1872-6771', '0166-218X']

DOI: https://doi.org/10.1016/j.dam.2021.07.002