arXiv subjects

Tomáš Masařík

Publications and source records attributed to Tomáš Masařík.

At least 19 recordsLinked to original sources

arXiv2026

Tree-independence number of $P_5$-free graphs with no large bicliques

The tree-independence number of a graph is the minimum, over all tree-decompositions of the graph, of the maximum size of an independent set contained in a bag. Graph classes of bounded tree-independence number have strong structural and algorithmic properties; however, the parameter can be unbounded even in quite restricted classes. In particular, the presence of an induced biclique $K_{\ell,\ell}$ forces tree-independence number at least $\ell$. This leads to the question whether large induced bicliques are the only obstruction to bounded tree-independence number in natural hereditary classes. A conjecture of Dallard, Krnc, Kwon, Milanič, Munaro, Štorgel, and Wiederrecht states that for all positive integers $t$ and $\ell$, ${\{P_t,K_{\ell,\ell}\}}$-free graphs have bounded tree-independence number. We prove this conjecture for ${t=5}$ by showing that every ${\{P_5,K_{\ell,\ell}\}}$-free graph has tree-independence number at most ${4\ell-4}$. We also obtain related bounds for the weaker parameter of $α$-degeneracy and answer a question of Hilaire, Milanič, and Vasić whether tree-independence number of ${\{P_5,K_{\ell,\ell}\}}$-free graphs exceeds $\ell$ by at most an additive constant.

math.CO↗

arXiv2026

Separator Theorem for Minor-Free Graphs in Linear Time

The planar separator theorem by Lipton and Tarjan [FOCS '77, SIAM Journal on Applied Mathematics '79] states that any planar graph with $n$ vertices has a balanced separator of size $O(\sqrt{n})$ that can be found in linear time. This landmark result kicked off decades of research on designing linear or nearly linear-time algorithms on planar graphs. In an attempt to generalize Lipton-Tarjan's theorem to nonplanar graphs, Alon, Seymour, and Thomas [STOC '90, Journal of the AMS '90] showed that any minor-free graph admits a balanced separator of size $O(\sqrt{n})$ that can be found in $O(n^{3/2})$ time. The superlinear running time in their separator theorem is a key bottleneck for generalizing algorithmic results from planar to minor-free graphs. Despite extensive research for more than two decades, finding a balanced separator of size $O(\sqrt{n})$ in (linear) $O(n)$ time for minor-free graphs remains a major open problem. Known algorithms either give a separator of size much larger than $O(\sqrt{n})$ or have superlinear running time, or both. In this paper, we answer the open problem affirmatively. Our algorithm is very simple: it runs a vertex-weighted variant of breadth-first search (BFS) a constant number of times on the input graph. Our key technical contribution is a weighting scheme on the vertices to guide the search for a balanced separator, offering a new connection between the size of a balanced separator and the existence of a clique-minor model. We believe that our weighting scheme may be of independent interest.

cs.DS↗

arXiv2026

Edge-decomposition into Two Triangular Forests is NP-complete

Let $\mathcal F$ be a graph class that is closed under topological minors and 1-sums, has decidable membership, contains a triangle, and is not the class of all graphs. Recently, Lee, Liu, and Tsai [ICALP 2026] showed that the edge-decomposition problem into $k \geq 3$ elements of $\mathcal F$ is NP-hard. In particular, their general hardness reduction covers a long-standing problem on outerthickness (when $\mathcal F$ is the class of outerplanar graphs). On the other hand, it is well known that decomposing a graph into forests is polynomial-time solvable, as implied by work of Edmonds [J. Res. Natl. Bur. Stand. B. 1965]. In this paper, we take a first step toward determining the complexity of edge-decomposition problems into just two graphs (the case $k=2$). We consider the simplest possible graph class $\mathcal F$ satisfying the criteria above: the triangular forests, that is, graphs in which every 2-connected component is a triangle. We prove that determining whether a graph can be edge-decomposed into two triangular forests is NP-complete.

cs.CC↗

arXiv2026

Induced Erdős--Pósa property for long holes, long thetas, and beyond

The induced Erdős--Pósa property in graphs relates the maximum number of pairwise anti-adjacent copies of an object with the minimum number of neighborhoods required to hit all copies. In this paper, the objects we consider are long cycles and long thetas, both as induced minors. Let $C_t$ denote the cycle with $t$ vertices and let $Θ_t$ be the graph consisting of three internally disjoint and anti-adjacent paths, each with $t$ internal vertices, connecting the same pair of distinct vertices. We show that for every fixed $t$, both $C_t$ and $Θ_t$ have the induced Erdős--Pósa property with respect to the induced minor relation. More precisely, for every integer $k$ and every graph $G$, one of the following two outcomes occurs: (i) $G$ contains $k$ pairwise vertex-disjoint and anti-adjacent copies of $C_t$ (resp., $Θ_t$) as induced minors, or (ii) there is a set $X \subseteq V(G)$ of size $\mathcal{O}(tk \log k)$ such that the set $N[X]$, consisting of $X$ and its neighbors, hits all $C_t$ (resp., all $Θ_t$) induced minors in $G$. This resolves in a strong form a special case of a conjecture of Ahn, Gollin, Huynh, and Kwon [SODA 2025]. From these results we derive that graphs that exclude $k$ disjoint copies of $Θ_t$ as an induced minor admit balanced separators consisting of the neighborhood of $\mathcal{O}(tk \log k)$ vertices. This in turn resolves a special case of a conjecture of Gartland and Lokshtanov and, combined with known techniques, yields a QPTAS for Maximum Weight Independent Set and a number of its generalizations.

math.CO↗

arXiv2026

General Strong Bound on the Uncrossed Number via a Tight Bound for the Maximum Uncrossed Subgraph Number

We investigate a very recent concept for visualizing various aspects of a graph in the plane using a collection of drawings introduced by Hliněný and Masařík [GD 2023]. Formally, given a graph $G$, we aim to find an uncrossed collection containing drawings of $G$ in the plane such that each edge of $G$ is not crossed in at least one drawing in the collection. The uncrossed number of $G$ ($unc(G)$) is the smallest integer $k$ such that an uncrossed collection for $G$ of size $k$ exists. The uncrossed number is lower-bounded by the well-known thickness, which is an edge-decomposition of $G$ into planar graphs. This connection gives a trivial lower-bound $\lceil\frac{|E(G)|}{3|V(G)|-6}\rceil \le unc(G)$. In a recent paper, Balko, Hliněný, Masařík, Orthaber, Vogtenhuber, and Wagner [GD 2024] presented the first non-trivial and general lower-bound on the uncrossed number. We summarize it in terms of dense graphs (where $|E(G)|=ε(|V(G)|)^2$ for some $ε>0$): $\lceil\frac{|E(G)|}{c_ε|V(G)|}\rceil \le unc(G)$, where $c_ε\ge 2.82$ is a constant depending on $ε$. We improve the lower-bound to state that $\lceil\frac{|E(G)|}{3|V(G)|-6-\sqrt{2|E(G)|}+\sqrt{6(|V(G)|-2)}}\rceil \le unc(G)$. Translated to dense graphs regime, the bound yields a multiplicative constant $c'_ε=3-\sqrt{(2-ε)}$ in the expression $\lceil\frac{|E(G)|}{c'_ε|V(G)|+o(|V(G)|)}\rceil \le unc(G)$. Hence, it is tight (up to low-order terms) for $ε\approx \frac{1}{2}$ as warranted by complete graphs. In fact, we formulate our result in the language of the maximum uncrossed subgraph number, that is, the maximum number of edges of $G$ that are not crossed in a drawing of $G$ in the plane. In that case, we also provide a construction certifying that our bound is asymptotically tight (up to lower-order terms) on dense graphs for all $ε>0$.

math.CO↗

arXiv2026

Fair Vertex Problems Parameterized by Cluster Vertex Deletion

In this paper we study fair variants of MSO$_1$ definable problems parameterized by cluster vertex deletion number, i.e., the smallest number of vertices required to be removed from the graph such that what remains is a collection of cliques. While typical graph problems seek the smallest set of vertices satisfying some property, their fair variants seek such a set that does not contain too many vertices in any neighborhood of any vertex. Formally, the task is to find a set $X\subseteq V(G)$ satisfying some MSO$_1$ definable property, whose fair cost is at most $k$, i.e., such that for all $v\in V(G)$ it holds that $|X\cap N(v)|\le k$. Recently, Knop, Masařík, and Toufar [MFCS 2019] showed that all fair MSO$_1$ definable problems can be solved in FPT time parameterized by the twin cover of a graph. They asked whether such a statement can be achieved for a more general parameterization by cluster vertex deletion number. In this paper, we prove that in full generality this is not possible by demonstrating W[1]-hardness. On the other hand, we give a sufficient condition under which a fair MSO$_1$ definable problem admits an FPT algorithm parameterized by the cluster vertex deletion number. Our algorithm is general enough to capture the fair variant of many natural graph problems such as the Fair Feedback Vertex Set problem, the Fair Vertex Cover problem, the Fair Dominating Set problem, the Fair Odd Cycle Transversal problem, as well as connected variants thereof. Moreover, we solve the Fair $[σ,ρ]$-Domination problem for $σ$ finite, or when both $σ$ and $ρ$ are cofinite. That is, given finite or cofinite $ρ,σ\subseteq \mathbb{N}$, the task is to find set of vertices $X\subseteq V(G)$ of fair cost at most $k$ such that for all $v\in X$, $|N(v)\cap X| \inσ$ and for all $v\in V(G)\setminus X$, $|N(v)\cap X|\inρ$.

cs.DS↗

arXiv2026

QPTAS for MWIS and finding large sparse induced subgraphs in graphs with few independent long holes

We present a quasipolynomial-time approximation scheme (QPTAS) for the Maximum Independent Set (\textsc{MWIS}) in graphs with a bounded number of pairwise vertex-disjoint and non-adjacent long induced cycles. More formally, for every fixed $s$ and $t$, we show a QPTAS for \textsc{MWIS} in graphs that exclude $sC_t$ as an induced minor. Combining this with known results, we obtain a QPTAS for the problem of finding a largest induced subgraph of bounded treewidth with given hereditary property definable in Counting Monadic Second Order Logic, in the same classes of graphs. This is a step towards a conjecture of Gartland and Lokshtanov which asserts that for any planar graph $H$, graphs that exclude $H$ as an induced minor admit a polynomial-time algorithm for the latter problem. This conjecture is notoriously open and even its weaker variants are confirmed only for very restricted graphs $H$.

cs.DS↗

arXiv2026

Optimal Path Partitions in Subcubic and Almost-subcubic Graphs

We consider the problem of partitioning the edges of a graph into as few paths as possible. This is a~subject of the classic conjecture of Gallai and a recurring topic in combinatorics. Regarding the complexity of partitioning a graph optimally, Peroché [Discret. Appl. Math., 1984] proved that it is NP-hard already on graphs of maximum degree four, even when we only ask if two paths suffice. We show that the problem is solvable in polynomial time on subcubic graphs and then we present an efficient algorithm for ``almost-subcubic'' graphs. Precisely, we prove that the problem is fixed-parameter tractable when parameterized by the edge-deletion distance to a subcubic graph. To this end, we reduce the task to model checking in first-order logic extended by disjoint-paths predicates ($\mathsf{FO}\text{+}\mathsf{DP}$) and then we employ the recent tractability result by Schirrmacher, Siebertz, Stamoulis, Thilikos, and Vigny [LICS 2024].

cs.DS↗

arXiv2025

Finding Diverse Solutions Parameterized by Cliquewidth

Finding a few solutions for a given problem that are diverse, as opposed to finding a single best solution to solve the problem, has recently become a notable topic in theoretical computer science. Recently, Baste, Fellows, Jaffke, Masařík, Oliveira, Philip, and Rosamond showed that under a standard structural parameterization by treewidth, one can find a set of diverse solutions for many problems with only a very small additional cost [Artificial Intelligence 2022]. In this paper, we investigate a much stronger graph parameter, the cliquewidth, which can additionally describe some dense graph classes. Broadly speaking, it describes graphs that can be recursively constructed by a few operations defined on graphs whose vertices are divided into a bounded number of groups while each such group behaves uniformly with respect to any operation. We show that for any vertex problem, if we are given a dynamic program solving that problem on cliquewidth decomposition, we can modify it to produce a few solutions that are as diverse as possible with as little overhead as in the above-mentioned treewidth paper. As a consequence, we prove that a diverse version of any MSO$_1$ expressible problem can be solved in linear FPT time parameterized by the cliquewidth, the number of sought solutions, and the number of quantifiers in the formula, which was a natural missing piece in the complexity landscape of structural graph parameters and logic for the diverse problems. We prove our results allowing for a more general natural collection of diversity functions compared to only two mostly studied diversity functions previously. That might be of independent interest as a larger pool of different diversity functions can highlight various aspects of different solutions to a problem.

cs.DS↗

arXiv2025

Maximum Weight Independent Set in Graphs with no Long Claws in Quasi-Polynomial Time

We show that the Maximum Weight Independent Set problem (MWIS) can be solved in quasi-polynomial time on $H$-free graphs (graphs excluding a fixed graph $H$ as an induced subgraph) for every $H$ whose every connected component is a path or a subdivided claw (i.e., a tree with at most three leaves). This completes the dichotomy of the complexity of MWIS in $\mathcal{F}$-free graphs for any finite set $\mathcal{F}$ of graphs into NP-hard cases and cases solvable in quasi-polynomial time, and corroborates the conjecture that the cases not known to be NP-hard are actually polynomial-time solvable. The key graph-theoretic ingredient in our result is as follows. Fix an integer $t \geq 1$. Let $S_{t,t,t}$ be the graph created from three paths on $t$ edges by identifying one endpoint of each path into a single vertex. We show that, given a graph $G$, one can in polynomial time find either an induced $S_{t,t,t}$ in $G$, or a balanced separator consisting of $\mathcal{O}(\log |V(G)|)$ vertex neighborhoods in $G$, or an extended strip decomposition of $G$ (a decomposition almost as useful for recursion for MWIS as a partition into connected components) with each particle of weight multiplicatively smaller than the weight of $G$. This is a strengthening of a result of Majewski, Masařík, Novotná, Okrasa, Pilipczuk, Rzążewski, and Sokołowski [ICALP 2022] which provided such an extended strip decomposition only after the deletion of $\mathcal{O}(\log |V(G)|)$ vertex neighborhoods. To reach the final result, we employ an involved branching strategy that relies on the structural lemma presented above.

cs.DS↗

arXiv2025

Minimizing an Uncrossed Collection of Drawings

In this paper, we introduce the following new concept in graph drawing. Our task is to find a small collection of drawings such that they all together satisfy some property that is useful for graph visualization. We propose investigating a property where each edge is not crossed in at least one drawing in the collection. We call such collection uncrossed. This property is motivated by a quintessential problem of the crossing number, where one asks for a drawing where the number of edge crossings is minimum. Indeed, if we are allowed to visualize only one drawing, then the one which minimizes the number of crossings is probably the neatest for the first orientation. However, a collection of drawings where each highlights a different aspect of a graph without any crossings could shed even more light on the graph's structure. We propose two definitions. First, the uncrossed number, minimizes the number of graph drawings in a collection, satisfying the uncrossed property. Second, the uncrossed crossing number, minimizes the total number of crossings in the collection that satisfy the uncrossed property. For both definitions, we establish initial results. We prove that the uncrossed crossing number is NP-hard, but there is an FPT algorithm parameterized by the solution size.

cs.CG↗

arXiv2025

Constricting the Computational Complexity Gap of the $4$-Coloring Problem in $(P_t,C_3)$-free Graphs

The $k$-Coloring problem on hereditary graph classes has been a deeply researched problem over the last decade. A hereditary graph class is characterized by a (possibly infinite) list of minimal forbidden induced subgraphs. We say that a graph is $(H_1,H_2,\ldots)$-free if it does not contain any of $H_1,H_2,\ldots$ as induced subgraphs. The complexity landscape of the problem remains unclear even when restricting to the case $k=4$ and classes defined by a few forbidden induced subgraphs. While the case of only one forbidden induced subgraph has been completely resolved lately, the complexity when considering two forbidden induced subgraphs still has a couple of unknown cases. In particular, $4$-Coloring on $(P_6,C_3)$-free graphs is polynomial while it is NP-hard on $(P_{22},C_3)$-free graphs. We provide a reduction showing NP-completeness of $4$-Coloring on $(P_t,C_3)$-free graphs for $19\leq t\leq 21$, thus constricting the gap of cases whose complexity remains unknown. Our proof includes a computer search ensuring that the graph family obtained through the reduction is indeed $P_{19}$-free.

cs.CC↗

arXiv2025

Path Eccentricity and Forbidden Induced Subgraphs

The path eccentricity of a connected graph $G$ is the minimum integer $k$ such that $G$ has a path such that every vertex is at distance at most $k$ from the path. A result of Duffus, Jacobson, and Gould from 1981 states that every connected $\{\text{claw}, \text{net}\}$-free graph $G$ has a Hamiltonian path, that is, $G$ has path eccentricity $0$. Several more recent works identified various classes of connected graphs with path eccentricity at most $1$, or, equivalently, graphs having a spanning caterpillar, including connected $P_5$-free graphs, AT-free graphs, and biconvex graphs. Generalizing all these results, we apply the work on structural distance domination of Bacsó and Tuza [Discrete Math., 2012] and characterize, for every positive integer $k$, graphs such that every connected induced subgraph has path eccentricity less than $k$. More specifically, we show that every connected $\{S_{k}, T_{k}\}$-free graph has a path eccentricity less than $k$, where $S_k$ and $T_k$ are two specific graphs of path eccentricity $k$ (a subdivided claw and the line graph of such a graph). As a consequence, every connected $H$-free graph has path eccentricity less than $k$ if and only if $H$ is an induced subgraph of $3P_{k}$ or $P_{2k+1} + P_{k-1}$. For such cases, we also provide a robust polynomial-time algorithm that finds a path witnessing the upper bound on the path eccentricity. Our main result also answers an open question of Bastide, Hilaire, and Robinson [Discrete Math., 2025].

math.CO↗

arXiv2025

Unbent Collections of Orthogonal Drawings

Recently, there has been interest in representing single graphs by multiple drawings; for example, using graph stories, storyplans, or uncrossed collections. In this paper, we apply this idea to orthogonal graph drawing. Due to the orthogonal drawing style, we focus on 4-graphs, that is, graphs of maximum degree 4. We restrict ourselves to plane graphs, that is, planar graphs whose embedding is fixed. Our goal is to represent any plane 4-graph $G$ by an unbent collection, that is, a collection of orthogonal drawings of $G$ that adhere to the embedding of $G$ and ensure that each edge of $G$ is drawn without bends in at least one of the drawings. We investigate two objectives. First, we consider minimizing the number of drawings in an unbent collection. We prove that every plane 4-graph can be represented by a collection with at most three drawings, which is tight. We also give necessary and sufficient conditions for a graph to admit an unbent collection of size $2$. Second, we consider minimizing the total number of bends over all drawings in an unbent collection. We show that this problem is NP-hard and give a 3-approximation algorithm. For the special case of plane triconnected cubic graphs, we show how to compute minimum-bend collections in linear time.

cs.CG↗

arXiv2025

On the Uncrossed Number of Graphs

Visualizing a graph $G$ in the plane nicely, for example, without crossings, is unfortunately not always possible. To address this problem, Masařík and Hliněný [GD 2023] recently asked for each edge of $G$ to be drawn without crossings while allowing multiple different drawings of $G$. More formally, a collection $\mathcal{D}$ of drawings of $G$ is uncrossed if, for each edge $e$ of $G$, there is a drawing in $\mathcal{D}$ such that $e$ is uncrossed. The uncrossed number $\mathrm{unc}(G)$ of $G$ is then the minimum number of drawings in some uncrossed collection of $G$. No exact values of the uncrossed numbers have been determined yet, not even for simple graph classes. In this paper, we provide the exact values for uncrossed numbers of complete and complete bipartite graphs, partly confirming and partly refuting a conjecture posed by Hliněný and Masařík. We also present a strong general lower bound on $\mathrm{unc}(G)$ in terms of the number of vertices and edges of $G$. Moreover, we prove NP-hardness of the related problem of determining the edge crossing number of a graph $G$, which is the smallest number of edges of $G$ taken over all drawings of $G$ that participate in a crossing. This problem was posed as open by Schaefer in his book [Crossing Numbers of Graphs 2018].

math.CO↗

arXiv2025

Random Embeddings of Graphs: The Expected Number of Faces in Most Graphs is Logarithmic

A random 2-cell embedding of a connected graph $G$ in some orientable surface is obtained by choosing a random local rotation around each vertex. Under this setup, the number of faces or the genus of the corresponding 2-cell embedding becomes a random variable. Random embeddings of two particular graph classes, those of a bouquet of $n$ loops and those of $n$ parallel edges connecting two vertices, have been extensively studied and are well-understood. However, little is known about more general graphs. The results of this paper explain why Monte Carlo methods cannot work for approximating the minimum genus of graphs. In his breakthrough work [Permutation-partition pairs, JCTB 1991], Stahl developed the foundation of "random topological graph theory". Most of his results have been unsurpassed until today. In our work, we analyze the expected number of faces of random embeddings (equivalently, the average genus) of a graph $G$. It was very recently shown that for any graph $G$, the expected number of faces is at most linear. We show that the actual expected number of faces $F(G)$ is almost always much smaller. In particular, we prove: 1) $\frac{1}{2}\ln n - 2 < \mathbb{E}[F(K_n)] \le 3.65 \ln n +o(1)$. 2) For random graphs $G(n,p)$ ($p=p(n)$), we have $\mathbb{E}[F(G(n,p))] \le \ln^2 n+\frac{1}{p}$. 3) For random models $B(n,Δ)$ containing only graphs, whose maximum degree is at most $Δ$, we obtain stronger bounds by showing that the expected number of faces is $Θ(\log n)$.

math.CO↗

arXiv2025

A Tight Meta-theorem for LOCAL Certification of MSO$_2$ Properties within Bounded Treewidth Graphs

Distributed networks are prone to errors so verifying their output is critical. Hence, we develop LOCAL certification protocols for graph properties in which nodes are given certificates that allow them to check whether their network as a whole satisfies some fixed property while only communicating with their local network. Most known LOCAL certification protocols are specifically tailored to the problem they work on and cannot be translated more generally. Thus we target general protocols that can certify any property expressible within a certain logical framework. We consider Monadic Second Order Logic (MSO$_2$), a powerful framework that can express properties such as non-$k$-colorability, Hamiltonicity, and $H$-minor-freeness. Unfortunately, in general, there are MSO$_2$-expressible properties that cannot be certified without huge certificates. For instance, non-3-colorability requires certificates of size $Ω(n^2/\log n)$ on general $n$-vertex graphs (Göös, Suomela 2016). Hence, we impose additional structural restrictions on the graph. We provide a LOCAL certification protocol for certifying any MSO$_2$-expressible property on graphs of bounded treewidth and, consequently, a LOCAL certification protocol for certifying bounded treewidth. That is for each integer $k$ and each MSO$_2$-expressible property $Π$ we give a LOCAL Certification protocol to certify that a graph satisfies $Π$ and has treewidth at most $k$ using certificates of size $\mathcal{O}(\log n)$ (which is asymptotically optimal). Our LOCAL certification protocol requires only one round of distributed communication, hence it is also proof-labeling scheme. Our result improves upon work by Fraigniaud, Montealegre, Rapaport, and Todinca (Algorithmica 2024), Bousquet, Feuilloley, Pierron (PODC 2022), and the very recent work of Baterisna and Chang.

cs.DC↗

arXiv2025

Single-conflict colorings of degenerate graphs

We consider the single-conflict coloring problem, a graph coloring problem in which each edge of a graph receives a forbidden ordered color pair. The task is to find a vertex coloring such that no two adjacent vertices receive a pair of colors forbidden at an edge joining them. We show that for any assignment of forbidden color pairs to the edges of a $d$-degenerate graph $G$ on $n$ vertices of edge-multiplicity at most $\log \log n$, $O(\sqrt{ d } \log n)$ colors are always enough to color the vertices of $G$ in a way that avoids every forbidden color pair. This answers a question of Dvořák, Esperet, Kang, and Ozeki for simple graphs (Journal of Graph Theory 2021).

math.CO↗