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Communication Dans Un Congrès Année : 2023

Monitoring edge-geodetic sets in graphs

Krishna Narayanan
Lekshmi Ramasubramony Sulochana

Résumé

We introduce a new graph-theoretic concept in the area of network monitoring. In this area, one wishes to monitor the vertices and/or the edges of a network (viewed as a graph) in order to detect and prevent failures. Inspired by two notions studied in the literature (edge-geodetic sets and distance-edge-monitoring sets), we define the notion of a monitoring edge-geodetic set (MEG-set for short) of a graph $G$ as an edge-geodetic set $S\subseteq V(G)$ of $G$ (that is, every edge of $G$ lies on some shortest path between two vertices of $S$) with the additional property that for every edge $e$ of $G$, there is a vertex pair $x, y$ of $S$ such that $e$ lies on \emph{all} shortest paths between $x$ and $y$. The motivation is that, if some edge $e$ is removed from the network (for example if it ceases to function), the monitoring probes $x$ and $y$ will detect the failure since the distance between them will increase. We explore the notion of MEG-sets by deriving the minimum size of a MEG-set for some basic graph classes (trees, cycles, unicyclic graphs, complete graphs, grids, hypercubes,...) and we prove an upper bound using the feedback edge set of the graph.
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Dates et versions

hal-03812622 , version 1 (12-10-2022)

Identifiants

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Florent Foucaud, Krishna Narayanan, Lekshmi Ramasubramony Sulochana. Monitoring edge-geodetic sets in graphs. 9th International Conference on Algorithms and Discrete Applied Mathematics (CALDAM 2023), Feb 2023, Gandhinagar, India. pp.245-256, ⟨10.1007/978-3-031-25211-2_19⟩. ⟨hal-03812622⟩
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