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DOI: 10.4230/LIPIcs.SAND.2022.9

Robustness of Distances and Diameter in a Fragile Network

Authors: Arnaud Casteigts, Timothée Corsini, Hervé Hocquard, and Arnaud Labourel

Published in: LIPIcs, Volume 221, 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022)

Abstract

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.

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Arnaud Casteigts, Timothée Corsini, Hervé Hocquard, and Arnaud Labourel. Robustness of Distances and Diameter in a Fragile Network. In 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 221, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{casteigts_et_al:LIPIcs.SAND.2022.9,
  author =	{Casteigts, Arnaud and Corsini, Timoth\'{e}e and Hocquard, Herv\'{e} and Labourel, Arnaud},
  title =	{{Robustness of Distances and Diameter in a Fragile Network}},
  booktitle =	{1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022)},
  pages =	{9:1--9:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-224-2},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{221},
  editor =	{Aspnes, James and Michail, Othon},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2022.9},
  URN =		{urn:nbn:de:0030-drops-159514},
  doi =		{10.4230/LIPIcs.SAND.2022.9},
  annote =	{Keywords: Dynamic networks, Longest path, Series-parallel graphs, Rooted minors}
}

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DOI: 10.4230/LIPIcs.IPEC.2019.15

Parameterized Complexity of Edge-Coloured and Signed Graph Homomorphism Problems

Authors: Florent Foucaud, Hervé Hocquard, Dimitri Lajou, Valia Mitsou, and Théo Pierron

Published in: LIPIcs, Volume 148, 14th International Symposium on Parameterized and Exact Computation (IPEC 2019)

Abstract

We study the complexity of graph modification problems with respect to homomorphism-based colouring properties of edge-coloured graphs. A homomorphism from an edge-coloured graph G to an edge-coloured graph H is a vertex-mapping from G to H that preserves adjacencies and edge-colours. We consider the property of having a homomorphism to a fixed edge-coloured graph H, which generalises the classic vertex-colourability property. The question we are interested in is the following: given an edge-coloured graph G, can we perform k graph operations so that the resulting graph admits a homomorphism to H? The operations we consider are vertex-deletion, edge-deletion and switching (an operation that permutes the colours of the edges incident to a given vertex). Switching plays an important role in the theory of signed graphs, that are 2-edge-coloured graphs whose colours are the signs + and -. We denote the corresponding problems (parameterized by k) by Vertex Deletion-H-Colouring, Edge Deletion-H-Colouring and Switching-H-Colouring. These problems generalise the extensively studied H-Colouring problem (where one has to decide if an input graph admits a homomorphism to a fixed target H). For 2-edge-coloured H, it is known that H-Colouring already captures the complexity of all fixed-target Constraint Satisfaction Problems. Our main focus is on the case where H is an edge-coloured graph of order at most 2, a case that is already interesting since it includes standard problems such as Vertex Cover, Odd Cycle Transversal and Edge Bipartization. For such a graph H, we give a PTime/NP-complete complexity dichotomy for all three Vertex Deletion-H-Colouring, Edge Deletion-H-Colouring and Switching-H-Colouring problems. Then, we address their parameterized complexity. We show that all Vertex Deletion-H-Colouring and Edge Deletion-H-Colouring problems for such H are FPT. This is in contrast with the fact that already for some H of order 3, unless PTime = NP, none of the three considered problems is in XP, since 3-Colouring is NP-complete. We show that the situation is different for Switching-H-Colouring: there are three 2-edge-coloured graphs H of order 2 for which Switching-H-Colouring is W[1]-hard, and assuming the ETH, admits no algorithm in time f(k)n^{o(k)} for inputs of size n and for any computable function f. For the other cases, Switching-H-Colouring is FPT.

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Florent Foucaud, Hervé Hocquard, Dimitri Lajou, Valia Mitsou, and Théo Pierron. Parameterized Complexity of Edge-Coloured and Signed Graph Homomorphism Problems. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 15:1-15:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{foucaud_et_al:LIPIcs.IPEC.2019.15,
  author =	{Foucaud, Florent and Hocquard, Herv\'{e} and Lajou, Dimitri and Mitsou, Valia and Pierron, Th\'{e}o},
  title =	{{Parameterized Complexity of Edge-Coloured and Signed Graph Homomorphism Problems}},
  booktitle =	{14th International Symposium on Parameterized and Exact Computation (IPEC 2019)},
  pages =	{15:1--15:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-129-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{148},
  editor =	{Jansen, Bart M. P. and Telle, Jan Arne},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.15},
  URN =		{urn:nbn:de:0030-drops-114765},
  doi =		{10.4230/LIPIcs.IPEC.2019.15},
  annote =	{Keywords: Graph homomorphism, Graph modification, Edge-coloured graph, Signed graph}
}

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Documents authored by Hocquard, Hervé

Robustness of Distances and Diameter in a Fragile Network

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Parameterized Complexity of Edge-Coloured and Signed Graph Homomorphism Problems

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