5 research outputs found
On the distance-edge-monitoring numbers of graphs
Foucaud et al. [Discrete Appl. Math. 319 (2022), 424-438] recently introduced
and initiated the study of a new graph-theoretic concept in the area of network
monitoring. For a set of vertices and an edge of a graph , let be the set of pairs with a vertex of and a vertex of
such that . For a vertex , let
be the set of edges such that there exists a vertex in with . A set of vertices of a graph is
distance-edge-monitoring set if every edge of is monitored by some
vertex of , that is, the set is nonempty. The
distance-edge-monitoring number of a graph , denoted by , is defined
as the smallest size of distance-edge-monitoring sets of . The vertices of
represent distance probes in a network modeled by ; when the edge
fails, the distance from to 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
\emph{distance-edge-monitoring sets}. In particular, we give upper and lower
bounds of , , , respectively, and extremal graphs
attaining the bounds are characterized. We also characterize the graphs with
Discovery of network properties with all-shortest-paths queries
We consider the problem of discovering properties (such as the diameter) of an unknown network G(V,E) with a minimum number of queries. Initially, only the vertex set V
of the network is known. Information about the edges and non-edges of the network can be obtained
by querying nodes of the network. A query at a node qâV returns the
union of all shortest paths from q to all other nodes in V. We study the
problem as an online problem - an algorithm does not initially know the
edge set of the network, and has to decide where to make the next query
based on the information that was gathered by previous queries. We
study how many queries are needed to discover the diameter, a minimal
dominating set, a maximal independent set, the minimum degree, and
the maximum degree of the network. We also study the problem of deciding with a minimum number of queries whether the network is 2-edge or
2-vertex connected. We use the usual competitive analysis to evaluate the
quality of online algorithms, i.e., we compare online algorithms with optimum offline algorithms. For all properties except maximal independent
set and 2-vertex connectivity we present and analyze online algorithms.
Furthermore we show, for all the aforementioned properties, that "many"
queries are needed in the worst case. As our query model delivers more
information about the network than the measurement heuristics that are
currently used in practise, these negative results suggest that a similar
behavior can be expected in realistic settings, or in more realistic models
derived from the all-shortest-paths query model
Discovery of network properties with all-shortest-paths queries
AbstractWe consider the problem of discovering properties (such as the diameter) of an unknown network G=(V,E) with a minimum number of queries. Initially, only the vertex set V of the network is known. Information about the edges and non-edges of the network can be obtained by querying nodes of the network. A query at a node qâV returns the union of all shortest paths from q to all other nodes in V. We study the problem as an online problemâan algorithm does not initially know the edge set of the network, and has to decide where to make the next query based on the information that was gathered by previous queries. We study how many queries are needed to discover the diameter, a minimal dominating set, a maximal independent set, the minimum degree, and the maximum degree of the network. We also study the problem of deciding with a minimum number of queries whether the network is 2-edge or 2-vertex connected. We use the usual competitive analysis to evaluate the quality of online algorithms, i.e., we compare online algorithms with the optimum offline algorithms. For all properties except the maximal independent set, 2-vertex connectivity and minimum/maximum degree, we present and analyze online algorithms. Furthermore we show, for all the aforementioned properties, that âmanyâ queries are needed in the worst case. As our query model delivers more information about the network than the measurement heuristics that are currently used in practice, these negative results suggest that a similar behavior can be expected in realistic settings, or in more realistic models derived from the all-shortest-paths query model