Searcharxiv⌕ Search

arXiv subjects

Hidde Koerts

Publications and source records attributed to Hidde Koerts.

12 recordsLinked to original sources
arXiv2026

Almost perfect graph classes

A graph $G$ is perfect if $\omega(H) = \chi(H)$ for each induced subgraph $H$ of $G$. In 2002, Chudnovsky, Robertson, Seymour, and Thomas famously proved the Strong Perfect Graph Theorem. Motivated by this forbidden induced subgraph characterization of the class of perfect graphs as well as the possible extension of efficient algorithms on perfect graphs, we consider the structure of graphs that are almost perfect. We say a graph is $c$-apex perfect if there is a constant $c$ number of vertices such that, upon the deletion of these vertices, what remains is a perfect graph. In this paper, we characterize the class of the sets of graphs $\mathcal{H}$ with $|\mathcal{H}|\leq 2$ for which there exists $c \in \mathbb{N}$ with the property that each $\mathcal{H}$-free graph is $c$-apex perfect. We also extend these results to several notable subclasses of perfect graphs, including chordal, interval, split, bipartite, and complete multipartite graphs.

math.CO↗
arXiv2026

Crossing tournaments are polynomially $\vec{\chi}$-bounded

Given a tournament $T$, Aboulker, Aubian, Charbit, and Lopes (2023) defined its clique number $\vec{\omega}(T)$ as the minimum clique number of a backedge graph of $T$, and raised the question: Which classes of tournaments are polynomially $\vec{\chi}$-bounded? Aboulker, Duron, Jacob, Kimbrough, Thomass\'{e}, and this work's authors (2026) showed that this holds for classes of tournaments whose arc sets may be written as the union of a bounded number of comparability digraphs. What about classes of tournaments that do not admit such a decomposition? The crossing tournaments of Nguyen, Scott, and Seymour (2025) are an example of such a class, as shown in the aforementioned 2026 work; we show that nonetheless crossing tournaments are polynomially $\vec{\chi}$-bounded by adapting a method of Davies and McCarty (2021) and Davies (2022). We additionally show that we cannot extend this result for crossing tournaments to tournaments with chordal graphs as backedge graphs.

math.CO↗
arXiv2026

Tree-independence number of $K_{1,d}$-free graph classes

In this paper, we investigate the tree-independence number of graph classes that do not contain $K_{1,d}$ as an induced subgraph. Dallard et al. conjectured that for any positive integer $d$ and any planar graph $H$, the class of all $K_{1,d}$-free graphs without $H$ as an induced minor has bounded tree-independence number. Our main contribution towards this conjecture is showing that the conjecture holds for outerstring graphs. Additionally we give linear and quadratic bounds for the tree-independence number of various $K_{1,d}$-free graph classes, sharpening previous bounds. Finally, we bound the tree-independence number of $K_{2,d}$-free graphs additionally forbidding holes of length at least $5$.

math.CO↗
arXiv2026

Decomposing tournaments into comparability graphs

In this note, we introduce the \emph{partial order decomposition number} of a digraph $D$, denoted $pod(D)$, defined as the minimum integer $k$ such that $A(D)=A(P_1)\cup\cdots\cup A(P_k)$, where $P_1,\ldots,P_k$ are partial orders on $V(D)$. We prove that $\dic(D)\le \diomega(D)^{pod(D)}$ for every digraph $D$. In particular, every class of digraphs with bounded $pod$ is polynomially $\dic$-bounded. We apply this to tournaments, showing that if $\mathcal C$ is a class of tournaments with bounded dichromatic number, then the closure of $\mathcal C$ under substitution is polynomially $\dic$-bounded, thereby making progress on a question of Aubian, Charbit, Lopes, and the first author. As further applications of $pod$, we prove that poset tournaments of bounded dimension are $\dic$-bounded, derive polynomial lower bounds on the directed clique number of an explicit family of tournaments, thereby answering a conjecture of Gutowski and Rams, and show that tournaments with bounded $pod$ have bounded domination number.

math.CO↗
arXiv2026

Characterizing Large Clique Number in Tournaments

Aboulker, Aubian, Charbit, and Lopes (2023) defined the clique number of a tournament to be the minimum clique number of one of its backedge graphs. Here we show that if $T$ is a tournament of sufficiently large clique number, then $T$ contains a subtournament of large clique number from one of two simple families of tournaments. In particular, large clique number is always certified by a bounded-size set. This answers a question of Aboulker, Aubian, Charbit, and Lopes (2023), and gives new insight into a line of research initiated by Kim and Kim (2018) into unavoidable subtournaments in tournaments with large dichromatic number.

math.CO↗
arXiv2026

Faster 3-colouring algorithm for graphs of diameter 3

We show that given an $n$-vertex graph $G$ of diameter 3 we can decide if $G$ is $3$-colourable in time $2^{O(n^{2/3-\varepsilon})}$ for any $\varepsilon < 1/33$. This improves on the previous best algorithm of $2^{O((n\log n)^{2/3})}$ from D\k{e}bski, Piecyk and Rz\k{a}\.zewski [Faster 3-coloring of small-diameter graphs, ESA 2021].

math.CO↗
arXiv2025

Intersections of graphs and $\chi$-boundedness

Given $k$ graphs $G_{1}, \ldots, G_{k}$, their intersection is the graph $(\cap_{i\in [k]}V(G_{i}), \cap_{i\in [k]}E(G_{i}))$. Given $k$ graph classes $\mathcal{G}_{1}, \ldots , \mathcal{G}_{k}$, we call the class $\{G: \forall i \in[k], \exists G_{i} \in \mathcal{G}_{i} \text{ such that } G=G_{1}\cap \ldots \cap G_{k}\}$ the graph-intersection of $\mathcal{G}_{1}, \ldots , \mathcal{G}_{k}$. The main motivation for the work presented in this paper is to try to understand under which conditions graph-intersection preserves $\chi$-boundedness. We consider the following two questions: Which graph classes have the property that their graph-intersection with every $\chi$-bounded class of graphs is $\chi$-bounded? We call such a class intersectionwise $\chi$-guarding. We prove that classes of graphs which admit a certain kind of decomposition are intersectionwise $\chi$-guarding. We provide necessary conditions that a finite set of graphs $\mathcal{H}$ should satisfy if the class of $\mathcal{H}$-free graphs is intersectionwise $\chi$-guarding, and we characterize the intersectionwise $\chi$-guarding classes which are defined by a single forbidden induced subgraph. Which graph classes have the property that, for every positive integer $k$, their $k$-fold graph-intersection is $\chi$-bounded? We call such a class intersectionwise self-$\chi$-guarding. We study intersectionwise self-$\chi$-guarding classes which are defined by a single forbidden induced subgraph, and we prove a result which allows us construct intersectionwise self-$\chi$-guarding classes from known intersectionwise $\chi$-guarding classes.

math.CO↗
arXiv2023

On the pre- and post-positional semi-random graph processes

We study the semi-random graph process, and a variant process recently suggested by Nick Wormald. We show that these two processes are asymptotically equally fast in constructing a semi-random graph $G$ that has property ${\mathcal P}$, for the following examples of ${\mathcal P}$: - ${\mathcal P}$ is the set of graphs containing a $d$-degenerate subgraph, where $d\ge 1$ is fixed; - ${\mathcal P}$ is the set of $k$-connected graphs, where $k\ge 1$ is fixed. In particular, our result of the $k$-connectedness above settles the open case $k=2$ of the original semi-random graph process. We also prove that there exist properties ${\mathcal P}$ where the two semi-random graph processes do not construct a graph in ${\mathcal P}$ asymptotically equally fast. We further propose some conjectures on ${\mathcal P}$ for which the two processes perform differently.

math.CO↗
arXiv2023

On heroes in digraphs with forbidden induced forests

We continue a line of research which studies which hereditary families of digraphs have bounded dichromatic number. For a class of digraphs $\mathcal{C}$, a hero in $\mathcal{C}$ is any digraph $H$ such that $H$-free digraphs in $\mathcal{C}$ have bounded dichromatic number. We show that if $F$ is an oriented star of degree at least five, the only heroes for the class of $F$-free digraphs are transitive tournaments. For oriented stars $F$ of degree exactly four, we show the only heroes in $F$-free digraphs are transitive tournaments, or possibly special joins of transitive tournaments. Aboulker et al. characterized the set of heroes of $\{H, K_{1} + \vec{P_{2}}\}$-free digraphs almost completely, and we show the same characterization for the class of $\{H, rK_{1} + \vec{P_{3}}\}$-free digraphs. Lastly, we show that if we forbid two "valid" orientations of brooms, then every transitive tournament is a hero for this class of digraphs.

math.CO↗
arXiv2022

On the $k$-independence number of graph products

The $k$-independence number of a graph, $α_k(G)$, is the maximum size of a set of vertices at pairwise distance greater than $k$, or alternatively, the independence number of the $k$-th power graph $G^k$. Although it is known that $α_k(G)=α(G^k)$, this, in general, does not hold for most graph products, and thus the existing bounds for $α$ of graph products cannot be used. In this paper we present sharp upper bounds for the $k$-independence number of several graph products. In particular, we focus on the Cartesian, tensor, strong, and lexicographic products. Some of the bounds previously known in the literature for $k=1$ follow as corollaries of our main results.

math.CO↗
arXiv2021

Near-Delaunay Metrics

We study metrics that assess how close a triangulation is to being a Delaunay triangulation, for use in contexts where a good triangulation is desired but constraints (e.g., maximum degree) prevent the use of the Delaunay triangulation itself. Our near-Delaunay metrics derive from common Delaunay properties and satisfy a basic set of design criteria, such as being invariant under similarity transformations. We compare the metrics, showing that each can make different judgments as to which triangulation is closer to Delaunay. We also present a preliminary experiment, showing how optimizing for these metrics under different constraints gives similar, but not necessarily identical results, on random and constructed small point sets.

cs.CG↗