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Rémi Watrigant

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6 papers
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6

MFCS Conference 2025 Conference Paper

Subcoloring of (Unit) Disk Graphs

  • Malory Marin
  • Rémi Watrigant

A subcoloring of a graph is a partition of its vertex set into subsets (called colors), each inducing a disjoint union of cliques. It is a natural generalization of the classical proper coloring, in which each color must instead induce an independent set. Similarly to proper coloring, we define the subchromatic number of a graph as the minimum integer k such that it admits a subcoloring with k colors, and the corresponding problem k-Subcoloring which asks whether a graph has subchromatic number at most k. In this paper, we initiate the study of the subcoloring of (unit) disk graphs. One motivation stems from the fact that disk graphs can be seen as a dense generalization of planar graphs where, intuitively, each vertex can be blown into a large clique-much like subcoloring generalizes proper coloring. Interestingly, it can be observed that every unit disk graph admits a subcoloring with at most 7 colors. We first prove that the subchromatic number can be 3-approximated in polynomial-time in unit disk graphs. We then present several hardness results for special cases of unit disk graphs which somehow prevents the use of classical approaches for improving this result. We show in particular that 2-Subcoloring remains NP-hard in triangle-free unit disk graphs, as well as in unit disk graphs representable within a strip of bounded height. We also solve an open question of Broersma, Fomin, Nešetřil, and Woeginger (2002) by proving that 3-Subcoloring remains NP-hard in co-comparability graphs (which contain unit disk graphs representable within a strip of height √3/2). Finally, we prove that every n-vertex disk graph admits a subcoloring with at most O(log³(n)) colors and present a O(log²(n))-approximation algorithm for computing the subchromatic number of such graphs. This is achieved by defining a decomposition and a special type of co-comparability disk graph, called Δ-disk graphs, which might be of independent interest.

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SODA Conference 2021 Conference Paper

Twin-width II: small classes

  • Édouard Bonnet
  • Colin Geniet
  • Eun Jung Kim 0002
  • Stéphan Thomassé
  • Rémi Watrigant

The recently introduced twin-width of a graph G is the minimum integer d such that G has a d-contraction sequence, that is, a sequence of | V ( G )| – 1 iterated vertex identifications for which the overall maximum number of red edges incident to a single vertex is at most d, where a red edge appears between two sets of identified vertices if they are not homogeneous in G (not fully adjacent nor fully non-adjacent). We show that if a graph admits a d -contraction sequence, then it also has a linear-arity tree of f ( d )-contractions, for some function f. Informally if we accept to worsen the twin-width bound, we can choose the next contraction from a set of Θ (| V ( G )|) pairwise disjoint pairs of vertices. This has two main consequences. First it permits to show that every bounded twin-width class is small, i. e. , has at most n! c n graphs labeled by [ n ], for some constant c. This unifies and extends the same result for bounded treewidth graphs [Beineke and Pippert, JCT '69], proper subclasses of permutations graphs [Marcus and Tardos, JCTA '04], and proper minor-free classes [Norine et al. , JCTB '06]. It implies in turn that bounded-degree graphs, interval graphs, and unit disk graphs have unbounded twin-width. The second consequence is an O (log n )-adjacency labeling scheme for bounded twin-width graphs, confirming several cases of the implicit graph conjecture. We then explore the small conjecture that, conversely, every small hereditary class has bounded twin-width. The conjecture passes many tests. Inspired by sorting networks of logarithmic depth, we show that log Θ (log log d ) n -subdivisions of K n (a small class when d is constant) have twin-width at most d. We obtain a rather sharp converse with a surprisingly direct proof: the log d +1 n -subdivision of K n has twin-width at least d. Secondly graphs with bounded stack or queue number (also small classes) have bounded twin-width. These sparse classes are surprisingly rich since they contain certain (small) classes of expanders. Thirdly we show that cubic expanders obtained by iterated random 2-lifts from K 4 [Bilu and Linial, Combinatorica '06] also have bounded twin-width. These graphs are related to so-called separable permutations and also form a small class. We suggest a promising connection between the small conjecture and group theory. Finally we define a robust notion of sparse twin-width. We show that for a hereditary class of bounded twin-width the five following conditions are equivalent: every graph in (1) is K t, t -fr ee for some fixed t, (2) has an adjacency matrix without a d-by-d division with a 1 entry in each d 2 cells for some fixed d, (3) has at most linearly many edges, (4) the subgraph closure of has bounded twin-width, and (5) has bounded expansion. We discuss how sparse classes with similar behavior with respect to clique subdivisions compare to bounded sparse twin-width.

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FOCS Conference 2020 Conference Paper

Twin-width I: tractable FO model checking

  • Édouard Bonnet
  • Eun Jung Kim 0002
  • Stéphan Thomassé
  • Rémi Watrigant

Inspired by a width invariant defined on permutations by Guillemot and Marx [SODA '14], we introduce the notion of twin-width on graphs and on matrices. Proper minor-closed classes, bounded rank-width graphs, map graphs, Kt - free unit d-dimensional ball graphs, posets with antichains of bounded size, and proper subclasses of dimension-2 posets all have bounded twin-width. On all these classes (except map graphs without geometric embedding) we show how to compute in polynomial time a sequence of d-contractions, witness that the twin-width is at most d. We show that FO model checking, that is deciding if a given first-order formula φ evaluates to true for a given binary structure G on a domain D, is FPT in |φ| on classes of bounded twin-width, provided the witness is given. More precisely, being given a d-contraction sequence for G, our algorithm runs in time f(d, |φ|)·|D| where f is a computable but non-elementary function. We also prove that bounded twin-width is preserved by FO interpretations and transductions (allowing operations such as squaring or complementing a graph). This unifies and significantly extends the knowledge on fixed-parameter tractability of FO model checking on non-monotone classes, such as the FPT algorithm on bounded-width posets by Gajarský et al. [FOCS '15].

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