Clément Rambaud

Quentin Claus, Gwenaël Joret, Clément Rambaud

We prove blow-up structure theorems for graphs excluding a tree or an apex-tree as a minor. First, we show that for every $t$-vertex tree $T$ with $t\geq 3$ and radius $h$, and every graph $G$ excluding $T$ as a minor, there exists a graph $H$ with pathwidth at most $2h-1$ such that $G$ is contained in $H\boxtimes K_{t-2}$ as a subgraph. This improves on a recent theorem of Dujmović, Hickingbotham, Joret, Micek, Morin, and Wood (2024), who proved the same result but with a larger bound on the order of the complete graph in the product. Second, we show that for every $t$-vertex tree $T$ with $t\geq 2$, radius $h$ and maximum degree $d$, and every graph $G$ excluding the apex-tree $T^+$ as a minor, where $T^+$ is the tree obtained by adding a universal vertex to $T$, there exists a graph $H$ with treewidth at most $4h-1$ such that $G$ is contained in $H\boxtimes K_{2(t-1)d}$. The bound on the treewidth of $H$ is best possible up to a factor $2$, and improves on a $2^{h+2}-4$ bound that follows from a recent result of Dujmović, Hickingbotham, Hodor, Joret, La, Micek, Morin, Rambaud, and Wood (2024).

Romain Bourneuf, Jędrzej Hodor, Piotr Micek et al.

Sample compression schemes were defined by Littlestone and Warmuth (1986) as an abstraction of the structure underlying many learning algorithms. In a sample compression scheme, we are given a large sample of vertices of a fixed hypergraph with labels indicating the containment in some hyperedge. The task is to compress the sample in such a way that we can retrieve the labels of the original sample. The size of a sample compression scheme is the amount of information that is kept in the compression. Every hypergraph with a sample compression scheme of bounded size must have bounded VC-dimension. Conversely, Moran and Yehudayoff (J. ACM, 2016) showed that every hypergraph of bounded VC-dimension admits a sample compression scheme of bounded size. We study a specific class of hypergraphs emerging from balls in graphs. The schemes that we construct (contrary to the ones constructed by Moran and Yehudayoff) are \textit{proper}, meaning that we retrieve not only the labeling of the original sample but also a hyperedge (ball) consistent with the original labeling. First, we prove that for every graph $G$ of treewidth at most $t$, the hypergraph of balls in $G$ has a proper sample compression scheme of size $\mathcal{O}(t\log t)$; this is tight up to the logarithmic factor and improves the quadratic (improper) bound that follows from the result of Moran and Yehudayoff. Second, we prove an analogous result for graphs of cliquewidth at most $t$.

Jędrzej Hodor, Hoang La, Piotr Micek et al.

Let $\mathcal{C}$ be a proper minor-closed class of graphs. Given the minors excluded in $\mathcal{C}$, we determine the maximum $q$-centered chromatic number and the maximum $q$th weak coloring number of graphs in $\mathcal{C}$ within an $\mathcal{O}(q)$-factor. Moreover, when $\mathcal{C}$ excludes a planar graph, we determine it within a constant factor. Our results imply that the $q$-centered chromatic number of $K_t$-minor-free graphs is in $\mathcal{O}(q^{t-1})$, improving on the previously known $\mathcal{O}(q^{h(t)})$ bound with a large and non-explicit function $h$. We include similar bounds for another family of parameters, the fractional treedepth fragility rates. All our bounds are proved via the same general framework.

Frédéric Havet, Clément Rambaud, Caroline Silva

In an oriented graph $\vec{G}$, the {\it inversion} of a subset $X$ of vertices consists in reversing the orientation of all arcs with both endvertices in $X$. The {\it $(\leq p)$-inversion graph} of a labelled graph $G$, denoted by ${\mathcal{I}}^{\leq p}(G)$, is the graph whose vertices are the labelled orientations of $G$ in which two labelled orientations $\vec{G}_1$ and $\vec{G}_2$ of $G$ are adjacent if and only if there is a set $X$ with $|X|\leq p$ whose inversion transforms $\vec{G}_1$ into $\vec{G}_2$. In this paper, we study the {\it $(\leq p)$-inversion diameter} of a graph, denoted by $\mathrm{id}^{\leq p}(G)$, which is the diameter of its $(\leq p)$-inversion graph. We show that there exists a smallest number $Ψ_p$ with $\frac{1}{4}p - \frac{3}{2} \leq Ψ_p \leq \frac{1}{2}p^2$ such that $\mathrm{id}^{\leq p}(G) \leq \left\lceil\frac{|E(G)|}{\lfloor p/2\rfloor}\right \rceil + Ψ_p$ for all graph $G$. We then establish better upper bounds for several families of graphs and in particular trees and planar graphs. Let us denote by $\mathrm{id}^{\leq p}_{\cal F}(n)$ (resp. $\mathrm{id}^{\leq p}_{\cal P}(n)$) the maximum $(\leq p)$-inversion diameter of a tree (resp. planar graph) of order $n$. For trees, we show $\mathrm{id}^{\leq 3}_{\cal F}(n) = \left\lceil \frac{n-1}{2}\right\rceil$, $\mathrm{id}^{\leq 4}_{\cal F}(n)=\frac{3}{8}n + Θ(1)$, $\mathrm{id}^{\leq 5}_{\cal F}(n)= \frac{2}{7}n + Θ(1)$, and $\mathrm{id}^{\leq p}_{\cal F}(n) \leq \frac{n-1}{p- c\sqrt{p}} + 2$ with $c = \sqrt{2 + \sqrt{2}}$ for all $p\geq 6$. For planar graphs, we prove $\mathrm{id}^{\leq 3}_{\cal P}(n) \leq \frac{11n}{6} - \frac{8}{3}$, $\mathrm{id}^{\leq 4}_{\cal P}(n) \leq \frac{4n}{3} + \frac{10}{3}$, and $\mathrm{id}^{\leq p}_{\cal P}(n) \leq \left\lceil\frac{3n-6}{\lfloor p/2\rfloor}\right \rceil + 8\lfloor p/2\rfloor - 8$ for all $p\geq 6$.