arXiv subjects

Petr Hliněný

Publications and source records attributed to Petr Hliněný.

At least 19 recordsLinked to original sources

arXiv2026

Measuring Depth of Matroids

Motivated by recently discovered connections between matroid depth measures and block-structured integer programming [ICALP 2020, 2022], we undertake a systematic study of recursive depth parameters for matrices and matroids, aiming to unify recently introduced and scattered concepts. We propose a general framework that naturally yields eight different depth measures for matroids, prove their fundamental properties and relationships, and relate them to two established notions in the field: matroid branch-depth and matroid tree-depth (a natural depth counterpart of matroid tree-width). In particular, we show that six of our eight measures are mutually functionally inequivalent, and among these, one is functionally equivalent to matroid branch-depth and another to matroid tree-depth. We also prove that all these depth measures, possibly except one, coincide on matroids and on matrices over any field, which, perhaps surprisingly, is not a trivial finding. Finally, we provide a comparison between the matroid parameters and classical depth measures of graphs.

math.CO↗

arXiv2026

k-Planar and Fan-Crossing Drawings and Transductions of Embeddable Graphs

We introduce, for every surface $Σ$, a two-way connection between definability of a graph class $\mathcal C$ by FO transductions (first-order logical transformations) of the graphs embeddable in $Σ$ and a certain variant of fan-crossing drawings of the graphs from $\mathcal C$ in $Σ$. If $\mathcal C$ is additionally of bounded maximum degree, then the restriction on drawings of the graphs from $\mathcal C$ in $Σ$ is simply to have a bounded number of crossings per edge (such as being $k$-planar for fixed~$k$ if $Σ$ is the plane). For graph classes, this connection allows us to derive non-transducibility results from the nonexistence of the said drawings and, conversely, from the nonexistence of a transduction to derive nonexistence of the said drawings. One example of such reasoning is as follows; since the class of 3D-grids is not transducible from the class of planar graphs, we can conclude that the class of 3D-grids is not $k$-planar for any fixed~$k$. On the other hand, the fact that the class of 3D-grids is not $k$-planar for any fixed~$k$ is known also via other means, and this conversely implies that the class of 3D-grids is not transducible from the class of planar graphs. We hope that this connection will help to draw a path to a possible proof that not all toroidal graphs are transducible from planar graphs. The result is based on a recent characterization of weakly sparse FO transductions of classes of bounded expansion by [Gajarský, Gładkowski, Jedelský, Pilipczuk and Toruńczyk, arXiv:2505.15655].

cs.CG↗

arXiv2026

Product Structure Meets Track Layouts

A track layout of a graph is a partition of its vertices into linearly ordered independent sets, called tracks, such that no two edges between the same pair of tracks cross. Given a graph, the goal in this context is to determine its track number, that is, the minimum number of tracks required for the graph to admit a track layout. In this work, we present upper bounds on the track number of graphs admitting a product structure. Our main contribution is an algorithm that computes a track layout with at most $(2h+1) \cdot r \cdot tn(H)$ tracks for every subgraph of the strong product $P^h \boxtimes K_r \boxtimes H$, where $P^h$ is the $h$-th power of a path $P$, $K_r$ is the complete graph on $r$ vertices, and $H$ is a graph with track number $tn(H)$. Combined with existing product-structure results from the literature, this algorithm yields upper bounds on the track number of several graph classes. For planar graphs, the obtained bound matches the current best-known upper bound of $225$. For $1$-planar and optimal $2$-planar graphs, our algorithm yields track layouts with at most $375$ tracks, while for genus-$k$, $k$-planar, $k$-framed, $k$-map, and $k$-string graphs it provides track layouts with a number of tracks that depends solely on $k$, thus establishing new upper bounds on the track number for these graph classes. The algorithm runs in linear time for planar graphs and, more generally, in $O(n + h \cdot r \cdot t + f_t(H))$ time whenever a corresponding product-structure decomposition of the input $n$-vertex graph is provided as part of the input, where $t=tn(H)$ and $f_t(H)$ is the time needed to compute a $t$-track layout of $H$. Furthermore, our algorithm only uses elementary linked-list data structures.

cs.DS↗

arXiv2026

Conflict-Free Coloring Planar Graphs with 4 Colors

We efficiently conflict-free color every planar graph with 4 colors. An (open-neighborhood) conflict-free coloring assigns colors to vertices in a way that every vertex v has a neighbor w such that the color of w is distinct from the colors of the other neighbors of v (i.e., the color of w is unique in the open neighborhood of v). A previous best upper bound on the conflict-free chromatic number of planar graphs was 5, and it is known that 4 colors are sometimes necessary. Deciding whether, e.g., a planar graph admits a conflict-free coloring with 3 colors is NP-complete. Our approach uses a refined variant of the classical Gallai-Edmonds decomposition and the Four Color Theorem. In fact, our result is equivalent to the Four Color Theorem.

math.CO↗

arXiv2026

Structure and generation of crossing-critical graphs

We study $c$-crossing-critical graphs, which are the minimal graphs that require at least $c$ edge-crossings when drawn in the plane. For $c=1$ there are only two such graphs without degree-2 vertices, $K_5$ and $K_{3,3}$, but for any fixed $c>1$ there exist infinitely many $c$-crossing-critical graphs. It has been previously shown that $c$-crossing-critical graphs have bounded path-width and contain only a bounded number of internally disjoint paths between any two vertices. We expand on these results, providing a more detailed description of the structure of crossing-critical graphs. On the way towards this description, we prove a new structural characterisation of plane graphs of bounded path-width. Then we show that every $c$-crossing-critical graph can be obtained from a $c$-crossing-critical graph of bounded size by replicating bounded-size parts that already appear in narrow "bands" or "fans" in the graph. This also gives an algorithm to generate all the $c$-crossing-critical graphs of at most given order $n$ in polynomial time per each generated graph.

math.CO↗

arXiv2026

Hereditary Graph Product Structure and $\cal H$-clique-width

We introduce H-clique-width, a new structural measure of graphs that aims to provide a hereditary analogue of the traditional graph product structure. The definition naturally generalises the ordinary clique-width concept. As a result, for a class H of graphs (such as the class of paths), the H-clique-width of a graph G equals the least integer t such that G is isomorphic to an induced subgraph of the strong product of a graph from H and a graph of clique-width t. We study basic properties of H-clique-width and compare it to other established structural parameters of graphs. Notably, we prove that the celebrated Planar graph product structure theorem by Dujmovic et al., and related graph product structure results, can all be formulated with the induced subgraph containment relation. In particular, every planar graph is isomorphic to an induced subgraph of the strong product of a path and a graph of tree-width 39.

math.CO↗

arXiv2026

A Unified FPT Framework for Crossing Number Problems

The basic (and traditional) crossing number problem is to determine the minimum number of crossings in a topological drawing of an input graph in the plane. We develop a unified framework yielding fixed-parameter tractable (FPT) algorithms for many generalized crossing number problems. Our framework takes the following form. We fix a surface S and a class D of "allowed" topological drawings of graphs in S (e.g., some class of drawings with at most t crossings). We assume that testing membership in D can be done algorithmically, and that restricting a drawing in D, extending it without adding any crossing, or transforming it with a self-homeomorphism of S yields a drawing that is also in D. Then deciding whether an input graph G has a drawing in D, and computing one if it is the case, is fixed-parameter tractable in (essentially) the genus of S and the maximum number of crossings in a drawing in D. More generally, we may take as input an edge-colored graph and distinguish crossings by the colors of the involved edges; and we may allow a bounded number of edge removals and vertex splits on G before drawing it. The proof is a reduction to the embeddability of a graph on a two-dimensional simplicial complex. This implies, in a unified way, linear or quadratic FPT algorithms for many topological crossing number variants established in the graph drawing community. Some of these variants already had previously published FPT algorithms, mostly relying on Courcelle's metatheorem, but our algorithms have a better runtime. Moreover, our framework extends to these crossing number variants in any fixed surface, and also allows us to fix the rotation system of the drawing of a graph in some variants.

cs.CG↗

arXiv2026

Crossing Number is NP-hard for Constant Path-width (and Tree-width)

The crossing number of a graph is the minimum number of edge crossings that a graph can have when drawn in the plane. Determining this number, known as the Crossing Number problem, is a celebrated problem in combinatorial optimization. It has been known to be NP-complete since the 1980s, and already showing its fixed-parameter tractability when parameterized by the vertex cover number required fairly involved techniques. In this paper, we prove that computing the crossing number exactly remains NP-hard even for graphs of path-width 12 (and as a result, for simple graphs of path-width 13 and tree-width 9). These results highlight that, although both path- and tree-decompositions have been highly successful tools in many graph algorithm scenarios, general crossing number computation is unlikely (under P $\neq$ NP) to be successfully tackled using graph decompositions of bounded width -- a question that had remained a 'tantalizing open problem' [S. Cabello, Hardness of Approximation for Crossing Number, 2013] till now.

cs.CG↗

arXiv2025

Computational Complexity of Covering Two-vertex Multigraphs with Semi-edges

We initiate the study of computational complexity of graph coverings, aka locally bijective graph homomorphisms, for {\em graphs with semi-edges}. The notion of graph covering is a discretization of coverings between surfaces or topological spaces, a notion well known and deeply studied in classical topology. Graph covers have found applications in discrete mathematics for constructing highly symmetric graphs, and in computer science in the theory of local computations. In 1991, Abello, Fellows, and Stillwell asked for a classification of the computational complexity of deciding if an input graph covers a fixed target graph, in the ordinary setting (of graphs with only edges). Although many general results are known, the full classification is still open. In spite of that, we propose to study the more general case of covering graphs composed of normal edges (including multiedges and loops) and so-called semi-edges. Semi-edges are becoming increasingly popular in modern topological graph theory, as well as in mathematical physics. They also naturally occur in the local computation setting, since they are lifted to matchings in the covering graph. We show some solvable cases and, in particular, completely characterize the complexity of the already very nontrivial problem of covering one- and two-vertex (multi)graphs with semi-edges. Our NP-hardness results are proven for simple input graphs, and in the case of regular two-vertex target graphs, even for bipartite ones. We remark that our new characterization results also strengthen previously known results for covering graphs without semi-edges, and they in turn apply to an infinite class of simple target graphs with at most two vertices of degree more than two. Some of the results are moreover proven in a more general setting (e.g., finding $k$-tuples of pairwise disjoint perfect matchings in regular graphs).

cs.DM↗

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

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

Transductions of Graph Classes Admitting Product Structure

In a quest to thoroughly understand the first-order transduction hierarchy of hereditary graph classes, some questions in particular stand out; such as, what properties hold for graph classes that are first-order transductions of planar graphs (and of similar classes)? When addressing this (so-far wide open) question, we turn to the concept of a product structure - being a subgraph of the strong product of a path and a graph of bounded tree-width, introduced by Dujmovic et al. [JACM 2020]. Namely, we prove that any graph class which is a first-order transduction of a class admitting such product structure, up to perturbations also meets a structural description generalizing the concept of a product structure in a dense hereditary way - the latter concept being introduced just recently by Hlineny and Jedelsky under the name of H-clique-width [MFCS 2024]. Using this characterization, we show that the class of the 3D grids, as well as a class of certain modifications of 2D grids, are not first-order transducible from classes admitting a product structure, and in particular not from the class of planar graphs.

cs.LO↗

arXiv2024

Stack and Queue Numbers of Graphs Revisited

A long-standing question of the mutual relation between the stack and queue numbers of a graph, explicitly emphasized by Dujmović and Wood in 2005, was "half-answered" by Dujmović, Eppstein, Hickingbotham, Morin and Wood in 2022; they proved the existence of a graph family with the queue number at most 4 but unbounded stack number. We give an alternative very short, and still elementary, proof of the same fact.

math.CO↗

arXiv2024

Note on Min-k-Planar Drawings of Graphs

The k-planar graphs, which are (usually with small values of k such as 1, 2, 3) subject to recent intense research, admit a drawing in which edges are allowed to cross, but each one edge is allowed to carry at most k crossings. In recently introduced [Binucci et al., GD 2023] min-k-planar drawings of graphs, edges may possibly carry more than k crossings, but in any two crossing edges, at least one of the two must have at most k crossings. In both concepts, one may consider general drawings or a popular restricted concept of drawings called simple. In a simple drawing, every two edges are allowed to cross at most once, and any two edges which share a vertex are forbidden to cross. While, regarding the former concept, it is for k<=3 known (but perhaps not widely known) that every general k-planar graph admits a simple k-planar drawing and this ceases to be true for any k>=4, the difference between general and simple drawings in the latter concept is more striking. We prove that there exist graphs with a min-2-planar drawing, or with a min-3-planar drawing avoiding crossings of adjacent edges, which have no simple min-k-planar drawings for arbitrarily large fixed k.

math.CO↗

arXiv2024

Complexity of Anchored Crossing Number and Crossing Number of Almost Planar Graphs

In this paper we deal with the problem of computing the exact crossing number of almost planar graphs and the closely related problem of computing the exact anchored crossing number of a pair of planar graphs. It was shown by [Cabello and Mohar, 2013] that both problems are NP-hard; although they required an unbounded number of high-degree vertices (in the first problem) or an unbounded number of anchors (in the second problem) to prove their result. Somehow surprisingly, only three vertices of degree greater than 3, or only three anchors, are sufficient to maintain hardness of these problems, as we prove here. The new result also improves the previous result on hardness of joint crossing number on surfaces by [Hliněný and Salazar, 2015]. Our result is best possible in the anchored case since the anchored crossing number of a pair of planar graphs with two anchors each is trivial, and close to being best possible in the almost planar case since the crossing number is efficiently computable for almost planar graphs of maximum degree 3 [Riskin 1996, Cabello and Mohar 2011].

cs.DM↗

arXiv2024

Twin-width of Planar Graphs; a Short Proof

The fascinating question of the maximum value of twin-width on planar graphs is nowadays not far from the final resolution; there is a lower bound of 7 coming from a construction by Král' and Lamaison [arXiv, September 2022], and an upper bound of 8 by Hliněný and Jedelský [arXiv, October 2022]. The upper bound (currently best) of 8, however, is rather complicated and involved. In the paper we give a short and simple self-contained proof that the twin-width of planar graphs is at most 11. We believe that this short proof can also shed more light on the topic of upper bound(s) on the twin-width of planar and beyond-planar graphs in general.

math.CO↗

arXiv2024

On 13-Crossing-Critical Graphs with Arbitrarily Large Degrees

A recent result of Bokal et al. [Combinatorica, 2022] proved that the exact minimum value of c such that c-crossing-critical graphs do not have bounded maximum degree is c=13. The key to that result is an inductive construction of a family of 13-crossing-critical graphs with many vertices of arbitrarily high degrees. While the inductive part of the construction is rather easy, it all relies on the fact that a certain 17-vertex base graph has the crossing number 13, which was originally verified only by a machine-readable computer proof. We provide a relatively short self-contained computer-free proof of the latter fact. Furthermore, we subsequently generalize the critical construction in order to provide a definitive answer to a remaining open question of this research area; we prove that for every c>=13 and integers d,q, there exists a c-crossing-critical graph with more than q vertices of each of the degrees 3,4,...,d.

math.CO↗

arXiv2023

Twin-width of Planar Graphs is at most 8, and some Related Bounds

Twin-width is a structural width parameter introduced by Bonnet, Kim, Thomassé and Watrigant [FOCS 2020], and has interesting applications in the areas of logic on graphs and in parameterized algorithmics. Very briefly, the essence of twin-width is in a gradual reduction (a contraction sequence) of the given graph down to a single vertex while maintaining limited difference of neighbourhoods of the vertices, and it can be seen as widely generalizing several other traditional structural parameters. While for many natural graph classes it is known that their twin-width is bounded, published upper bounds on the twin-width in non-trivial cases are very often "astronomically large". We focus on planar graphs, which are known to have bounded twin-width already since the introduction of it, but it took some time for the first explicit "non-astronomical" upper bounds to come. Namely, in the order of preprint appearance, it was the bound of at most 183 by Jacob and Pilipczuk [arXiv, January 2022], and 583 by Bonnet, Kwon and Wood [arXiv, February 2022]. Subsequent arXiv manuscripts in 2022 improved the bound down to 37 (Bekos et al.), 11 and 9 (both by Hliněný). We further elaborate on the approach used in the latter manuscripts, proving that the twin-width of every planar graph is at most 8, and construct a witnessing contraction sequence in linear time. Note that the currently best lower-bound planar example is of twin-width 7, by Král and Lamaison [arXiv, September 2022]. We also prove small explicit upper bounds on the twin-width of bipartite planar and 1-planar graphs (6 and 16), and of map graphs (38). The common denominator of all these results is the use of a novel specially crafted recursive decomposition of planar graphs, which may be found useful also in other areas.

math.CO↗