Robustness of Distances and Diameter in a Fragile Network
A property of a graph G is robust if it remains unchanged in all connected spanning subgraphs of G. This form of robustness is motivated by networking contexts where some links eventually fail permanently, and the network keeps being used so long as it is connected. It is then natural to ask how certain properties of the network may be impacted as the network deteriorates. In this paper, we focus on two particular properties, which are the diameter, and pairwise distances among nodes. Surprisingly, the complexities of deciding whether these properties are robust are quite different: deciding the robustness of the diameter is coNP-complete, whereas deciding the robustness of the distance between two given nodes has a linear time complexity. This is counterintuitive, because the diameter consists of the maximum distance over all pairs of nodes, thus one may expect that the robustness of the diameter reduces to testing the robustness of pairwise distances. On the technical side, the difficulty of the diameter is established through a reduction from hamiltonian paths. The linear time algorithm for deciding robustness of the distance relies on a new characterization of two-terminal series-parallel graphs (TTSPs) in terms of excluded rooted minor, which may be of independent interest.
Dynamic networks
Longest path
Series-parallel graphs
Rooted minors
Mathematics of computing~Paths and connectivity problems
Networks~Network dynamics
Theory of computation~Complexity classes
9:1-9:16
Regular Paper
ANR project ESTATE (ANR-16-CE25-0009-03) and DREAMY (ANR-21-CE48-0003)
Arnaud
Casteigts
Arnaud Casteigts
LaBRI, CNRS, Université de Bordeaux, Bordeaux INP, France
https://www.labri.fr/perso/acasteig/
https://orcid.org/0000-0002-7819-7013
Timothée
Corsini
Timothée Corsini
LaBRI, CNRS, Université de Bordeaux, Bordeaux INP, France
https://www.labri.fr/perso/tcorsini/
https://orcid.org/0000-0003-1055-5627
Hervé
Hocquard
Hervé Hocquard
LaBRI, CNRS, Université de Bordeaux, Bordeaux INP, France
https://www.labri.fr/perso/hocquard/
https://orcid.org/0000-0001-8194-4684
Arnaud
Labourel
Arnaud Labourel
Aix Marseille Univ, CNRS, LIS, Marseille, France
http://pageperso.lif.univ-mrs.fr/~arnaud.labourel
https://orcid.org/0000-0003-0162-1899
10.4230/LIPIcs.SAND.2022.9
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Arnaud Casteigts, Timothée Corsini, Hervé Hocquard, and Arnaud Labourel
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