1. Computing tree decompositions with small independence numberClément Jean Dallard, Fedor V. Fomin, Petr A. Golovach, Tuukka Korhonen, Martin Milanič, 2026, original scientific article Abstract: The independence number of a tree decomposition is the maximum of the independence numbers of the subgraphs induced by its bags. The tree-independence number of a graph is the minimum independence number of a tree decomposition of it. Several NP-hard graph problems, like maximum-weight independent set, can be solved in time n^O(k) if the input n-vertex graph is given together with a tree decomposition of independence number k. Yolov, in SODA 2018, gave an algorithm that, given an n-vertex graph G and an integer k, in time n^O(k^3) either constructs a tree decomposition of G whose independence number is O(k^3) or correctly reports that the tree-independence number of G is larger than k. In this article, we first give an algorithm for computing the tree-independence number with a better approximation ratio and running time and then prove that our algorithm is, in some sense, the best one can hope for. Our second result is that the exact computation of the tree-independence number is para-NP-hard: We show that for every constant k ≥ 4 it is NP-complete to decide whether a given graph has the tree-independence number at most k.
Keywords: tree-independence number, approximation, parameterized algorithm
Published in RUP: 16.12.2025; Views: 237; Downloads: 3
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2. Induced minor models : Structural properties and algorithmic consequencesNicolas Bousquet, Clément Jean Dallard, Maël Dumas, Claire Hilaire, Martin Milanič, Anthony Perez, Nicolas Trotignon, 2026, original scientific article Abstract: A graph H is an induced minor of G if there exists an induced minor model of H in G, that is, a collection of pairwise disjoint subsets of vertices of G labeled by the vertices of H, each inducing a connected subgraph in G, such that two vertices of H are adjacent if and only if there is an edge in G between the corresponding subsets. In this paper, we investigate structural properties of induced minor models, including bounds on treewidth and chromatic number of the subgraphs induced by minimal induced minor models. As algorithmic applications of our structural results, we make use of recent developments regarding tree-independence number to show that if H is the 4-wheel, the 5-vertex complete graph minus an edge, or a complete bipartite graph K2,q , then there is a polynomial-time algorithm to find in a given graph G an induced minor model of H in G, if there is one. We also develop an alternative polynomial-time algorithm for recognizing graphs that do not contain K2,3 as an induced minor, which revolves around the idea of detecting the induced subgraphs whose presence is forced when the input graph contains K2,3 as an induced minor. It turns out that all these induced subgraphs are Truemper configurations.
Keywords: induced minor, treewidth, chromatic number, tree-independence number, Truemper configuration
Published in RUP: 16.12.2025; Views: 203; Downloads: 3
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5. Allocating indivisible items with minimum dissatisfaction on preference graphsNina Chiarelli, Clément Jean Dallard, Andreas Darmann, Stefan Lendl, Martin Milanič, Peter Muršič, Nevena Pivač, Ulrich Pferschy, 2021, published scientific conference contribution Keywords: fair division, partial order, preference graph
Published in RUP: 29.10.2021; Views: 3955; Downloads: 24
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6. Bonsma, Paul; Paulusma, Daniël: Using contracted solution graphs for solving reconfiguration problems. (English summary) Acta Inform. 56 (2019), no. 7-8, 619-648.Clément Jean Dallard, 2021, review, book review, critique Keywords: reconfiguration, dynamic programming, graph coloring
Published in RUP: 26.10.2021; Views: 3553; Downloads: 12
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9. On girth and the parameterized complexity of token sliding and token jumpingValentin Bartier, Nicolas Bousquet, Clément Jean Dallard, Kyle Lomer, Amer E. Mouawad, 2021, original scientific article Keywords: combinatorial reconfiguration, independent set, token jumping, token sliding, parameterized complexity
Published in RUP: 16.07.2021; Views: 2361; Downloads: 31
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