Colouring exact distance graphs of chordal graphs

For a graph $G=(V,E)$ and positive integer $p$, the exact distance-$p$ graph $G^{[\natural p]}$ is the graph with vertex set $V$ and with an edge between vertices $x$ and $y$ if and only if $x$ and $y$ have distance $p$. Recently, there has been an effort to obtain bounds on the chromatic number $\chi(G^{[\natural p]})$ of exact distance-$p$ graphs for $G$ from certain classes of graphs. In particular, if a graph $G$ has tree-width $t$, it has been shown that $\chi(G^{[\natural p]}) \in \mathcal{O}(p^{t-1})$ for odd $p$, and $\chi(G^{[\natural p]}) \in \mathcal{O}(p^{t}\Delta(G))$ for even $p$. We show that if $G$ is chordal and has tree-width $t$, then $\chi(G^{[\natural p]}) \in \mathcal{O}(p\, t^2)$ for odd $p$, and $\chi(G^{[\natural p]}) \in \mathcal{O}(p\, t^2 \Delta(G))$ for even $p$. If we could show that for every graph $H$ of tree-width $t$ there is a chordal graph $G$ of tree-width $t$ which contains $H$ as an isometric subgraph (i.e., a distance preserving subgraph), then our results would extend to all graphs of tree-width $t$. While we cannot do this, we show that for every graph $H$ of genus $g$ there is a graph $G$ which is a triangulation of genus $g$ and contains $H$ as an isometric subgraph.Comment: 11 pages, 2 figures. Versions 2 and 3 include minor changes, which arise from reviewers' comment

Improved Bounds for the Excluded-Minor Approximation of Treedepth

Treedepth, a more restrictive graph width parameter than treewidth and pathwidth, plays a major role in the theory of sparse graph classes. We show that there exists a constant C such that for every integers a,b >= 2 and a graph G, if the treedepth of G is at least Cab log a, then the treewidth of G is at least a or G contains a subcubic (i.e., of maximum degree at most 3) tree of treedepth at least b as a subgraph. As a direct corollary, we obtain that every graph of treedepth Omega(k^3 log k) is either of treewidth at least k, contains a subdivision of full binary tree of depth k, or contains a path of length 2^k. This improves the bound of Omega(k^5 log^2 k) of Kawarabayashi and Rossman [SODA 2018]. We also show an application for approximation algorithms of treedepth: given a graph G of treedepth k and treewidth t, one can in polynomial time compute a treedepth decomposition of G of width O(kt log^{3/2} t). This improves upon a bound of O(kt^2 log t) stemming from a tradeoff between known results. The main technical ingredient in our result is a proof that every tree of treedepth d contains a subcubic subtree of treedepth at least d * log_3 ((1+sqrt{5})/2)

Solving Connectivity Problems Parameterized by Treedepth in Single-Exponential Time and Polynomial Space

A breakthrough result of Cygan et al. (FOCS 2011) showed that connectivity problems parameterized by treewidth can be solved much faster than the previously best known time ?^*(2^{?(twlog tw)}). Using their inspired Cut&Count technique, they obtained ?^*(?^tw) time algorithms for many such problems. Moreover, they proved these running times to be optimal assuming the Strong Exponential-Time Hypothesis. Unfortunately, like other dynamic programming algorithms on tree decompositions, these algorithms also require exponential space, and this is widely believed to be unavoidable. In contrast, for the slightly larger parameter called treedepth, there are already several examples of matching the time bounds obtained for treewidth, but using only polynomial space. Nevertheless, this has remained open for connectivity problems. In the present work, we close this knowledge gap by applying the Cut&Count technique to graphs of small treedepth. While the general idea is unchanged, we have to design novel procedures for counting consistently cut solution candidates using only polynomial space. Concretely, we obtain time ?^*(3^d) and polynomial space for Connected Vertex Cover, Feedback Vertex Set, and Steiner Tree on graphs of treedepth d. Similarly, we obtain time ?^*(4^d) and polynomial space for Connected Dominating Set and Connected Odd Cycle Transversal

Decomposition horizons: from graph sparsity to model-theoretic dividing lines

Let $\mathscr C$ be a hereditary class of graphs. Assume that for every $p$ there is a hereditary NIP class $\mathscr D_p$ with the property that the vertex set of every graph $G\in\mathscr C$ can be partitioned into $N_p=N_p(G)$ parts in such a way that the union of any $p$ parts induce a subgraph in $\mathscr D_p$ and $\log N_p(G)\in o(\log |G|)$. We prove that $\mathscr C$ is (monadically) NIP. Similarly, if every $\mathscr D_p$ is stable, then $\mathscr C$ is (monadically) stable. Results of this type lead to the definition of decomposition horizons as closure operators. We establish some of their basic properties and provide several further examples of decomposition horizons

A Cubic Vertex-Kernel for Trivially Perfect Editing

International audienc

Modulo-Counting First-Order Logic on Bounded Expansion Classes

We prove that, on bounded expansion classes, every first-order formula with modulo counting is equivalent, in a linear-time computable monadic lift, to an existential first-order formula. As a consequence, we derive, on bounded expansion classes, that first-order transductions with modulo counting have the same encoding power as existential first-order transductions. Also, modulo-counting first-order model checking and computation of the size of sets definable in modulo-counting first-order logic can be achieved in linear time on bounded expansion classes. As an application, we prove that a class has structurally bounded expansion if and only if is a class of bounded depth vertex-minors of graphs in a bounded expansion class. We also show how our results can be used to implement fast matrix calculus on bounded expansion matrices over a finite field.Comment: submitted to CSGT2022 special issu