Monitoring the edges of a graph using distances

Foucaud, Florent; Kao, Shih-Shun; Klasing, Ralf; Miller, Mirka; Ryan, Joe

doi:10.1016/j.dam.2021.07.002

Cited by 19 publications

(42 citation statements)

References 18 publications

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“…This is an improvement over the results from the conference version[12], where we proved dem(G) ≤ 5 fes(G)−5.…”

contrasting

confidence: 46%

“…We will now give a weaker (but similar) bound for any value of fes(G). This improves on our previous bound from [12]. A first candidate would be to select the set C of all core vertices of G b .…”

Section: Connection To Feedback Edge Set Numbermentioning

confidence: 81%

“…* 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.† A preliminary version of this paper appeared as [12].‡ The authors acknowledge the financial support from the ANR project HOSIGRA (ANR-17-CE40-0022), the IFCAM project "Applications of graph homomorphisms" (MA/IFCAM/18/39), and the Programme IdEx Bordeaux -SysNum…”

mentioning

confidence: 99%

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Monitoring the Edges of a Graph Using Distances

Foucaud

Klasing

Miller

et al. 2020

Lecture Notes in Computer Science

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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 dem(G) is lowerbounded 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. * 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. † A preliminary version of this paper appeared as [12]. ‡ The authors acknowledge the financial support from the ANR project HOSIGRA (ANR-17-CE40-0022), the IFCAM project "Applications of graph homomorphisms" (MA/IFCAM/18/39), and the Programme IdEx Bordeaux-SysNum (ANR-10-IDEX-03-02).

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“…This is an improvement over the results from the conference version[12], where we proved dem(G) ≤ 5 fes(G)−5.…”

contrasting

confidence: 46%

Section: Connection To Feedback Edge Set Numbermentioning

confidence: 81%

mentioning

confidence: 99%

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Monitoring the Edges of a Graph Using Distances

Foucaud

Klasing

Miller

et al. 2020

Lecture Notes in Computer Science

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“…Foucaud et al [8] showed that 1 ≤ dem(G) ≤ n − 1 for any G with order n, and characterized graphs with dem(G) = 1, 2, n − 1.…”

Section: Preliminariesmentioning

confidence: 99%

“…Foucaud et al [8] recently introduced a new concept of network monitoring using distance probes, called distance-edge-monitoring. Networks are naturally modeled by finite undirected simple connected graphs, whose vertices represent computers and whose edges represent connections between them.…”

Section: Introductionmentioning

confidence: 99%

On the distance-edge-monitoring numbers of graphs

Yang,

Klasing,

Mao

et al. 2022

Preprint

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Foucaud et al. [Discrete Appl. Math. 319 (2022), [424][425][426][427][428][429][430][431][432][433][434][435][436][437][438] recently introduced and initiated the study of a new graph-theoretic concept in the area of network monitoring. For a set M of vertices and an edge e of a graph G, let P (M, e) be the set of pairs (x, y) with a vertex x of M and a vertex y of V (G) such that d G (x, y) = d G−e (x, y). For a vertex x, let EM (x) be the set of edges e such that there exists a vertex v in G with (x, v) ∈ P ({x}, e). A set M of vertices of a graph G is distance-edge-monitoring set if every edge e of G is monitored by some vertex of M , that is, the set P (M, e) is nonempty. The distance-edge-monitoring number of a graph G, denoted by dem(G), is defined as the smallest size of distance-edge-monitoring sets of 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 continue the study of distanceedge-monitoring sets. In particular, we give upper and lower bounds of P (M, e), EM (x), dem(G), respectively, and extremal graphs attaining the bounds are characterized. We also characterize the graphs with dem(G) = 3.

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Monitoring Edge-Geodetic Sets in Graphs

Foucaud

2023

Lecture Notes in Computer Science

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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 Ď V pGq 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 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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Monitoring the edges of a graph using distances

Cited by 19 publications

References 18 publications

Monitoring the Edges of a Graph Using Distances

Monitoring the Edges of a Graph Using Distances

On the distance-edge-monitoring numbers of graphs

Monitoring Edge-Geodetic Sets in Graphs

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