arXiv subjects

Jan Kratochvíl

Publications and source records attributed to Jan Kratochvíl.

At least 19 recordsLinked to original sources

arXiv2026

How Close is a Tree to a Euclidean Minimum Spanning Tree?

Let $Γ$ be a straight-line crossing-free drawing of a tree $T$. A \emph{bad pair} in $Γ$ is a pair of non-adjacent vertices of $T$ whose Euclidean distance in $Γ$ is smaller than the length of the longest edge in the path connecting them in~$Γ$. When $Γ$ has no bad pairs, $Γ$ is a Euclidean Minimum Spanning Tree of its vertex set (or EMST-drawing for short). Deciding whether a tree of maximum degree at most six admits an EMST-drawing is known to be \NP-hard. In contrast, we characterize those caterpillars that admit an EMST-drawing. The characterization gives rise to a linear-time algorithm that decides if a caterpillar admits an EMST-drawing, and in the affirmative case, computes such a drawing. For caterpillars of maximum degree six, we further present a linear-time algorithm to compute a crossing-free straight-line drawing with the minimum number of bad pairs. For $n$-vertex trees with maximum vertex degree $Δ$, we prove the $Δ^2n\log n$ upper bound on the minimum number of bad pairs. In the special case of stars, we construct a drawing with the minimum number of bad pairs.

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arXiv2026

Two Results on Outer-String Graphs

An \emph{outer-string representation} of a graph $G$ is an intersection representation of $G$ where vertices are represented by curves (strings) inside the unit disk and each curve has exactly one endpoint on the boundary of the unit disk (the anchor of the curve). Additionally, if each two curves are allowed to cross at most once, we call this an \emph{outer-$1$-string representation} of $G$. If we impose a cyclic ordering on the vertices of $G$ and require the cyclic order of the anchors to respect this cyclic order, such a representation is called a \emph{constrained outer-string representation}. In this paper, we present two results about graphs admitting outer-string representations. Firstly, we show that for a bipartite graph $G$ (and, more generally, for any $\{C_3,C_5\}$-free graph $G$) with a given cyclic order of vertices, we can decide in polynomial time whether $G$ admits a constrained outer-string representation. Our algorithm follows from a characterization by a single forbidden configuration, similar to that of Biedl et al. [GD 2024] for chordal graphs. Secondly, we answer an open question from the same authors and show that determining whether a given graph admits an outer-1-string representation is NP-hard. More generally, we show that it is NP-hard to determine if a given graph $G$ admits an outer-$k$-string representation for any fixed $k\ge1$.

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arXiv2026

Path Cover, Hamiltonicity, and Independence Number: An FPT Perspective

The classic theorem of Gallai and Milgram (1960) generalizes several fundamental results in Graph Theory, such as Dilworth's theorem on posets and Kőnig's theorem on matchings in bipartite graphs. The theorem asserts that for every graph G, the vertex set of G can be partitioned into at most α(G) vertex-disjoint paths, where α(G) is the maximum size of an independent set in G. The proof of the Gallai-Milgram theorem is constructive and yields a polynomial-time algorithm that computes a covering of G by at most α(G) vertex-disjoint paths. While the Gallai-Milgram theorem is tight, it was not known prior to our work whether deciding if a graph G could be covered by fewer than α(G) vertex-disjoint paths can be done in polynomial time. We resolve this question by proving the following algorithmic extension of the Gallai-Milgram theorem for undirected graphs: There is an algorithm that, for an n-vertex graph G and an integer parameter k \ge 1, runs in time 2^{2^{O(k^4\log{k})}} \cdot n^{O(1)} and outputs a path cover P of G together with - a correct conclusion that P is a minimum-size path cover, or - an independent set of size |P|+k, certifying that P contains at most α(G) - k paths. The proof of our algorithmic extension of the Gallai-Milgram theorem is non-trivial and builds on several novel algorithmic ideas. One of the key subroutines in our algorithm is an FPT algorithm, parameterized by α(G), for deciding whether G contains a Hamiltonian path. This result is of independent interest - prior to our work no polynomial-time algorithm for deciding Hamiltonicity was known even for graphs with independence number at most three. Moreover, the algorithmic techniques we develop apply to a wide array of problems in undirected graphs, including Hamiltonian Cycle, Path Cover, Largest Linkage, and Topological Minor Containment.

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arXiv2026

Grounded String Representations of Series-Parallel Graphs without Transitive Edges

In a {\em grounded string representation} of a graph there is a horizontal line $\ell$ and each vertex is represented as a simple curve below $\ell$ with one end point on $\ell$ such that two curves intersect if and only if the respective vertices are adjacent. A grounded string representation is a {\em grounded L-reverseL-representation} if each vertex is represented by a 1-bend orthogonal polyline. It is a {\em grounded L-representation} if in addition all curves are L-shaped. We show that every biconnected series-parallel graph without edges between the two vertices of a separation pair (i.e., {\em transitive edges}) admits a grounded L-reverseL-representation if and only if it admits a grounded string representation. Moreover, we can test in linear time whether such a representation exists. We also construct a biconnected series-parallel graph without transitive edges that admits a grounded L-reverseL-representation, but no grounded L-representation.

cs.CG↗

arXiv2025

Edge-Constrained Hamiltonian Paths on a Point Set

Let S be a set of distinct points in general position in the Euclidean plane. A plane Hamiltonian path on S is a crossing-free geometric path such that every point of S is a vertex of the path. It is known that, if S is sufficiently large, there exist three edge-disjoint plane Hamiltonian paths on S. In this paper we study an edge-constrained version of the problem of finding Hamiltonian paths on a point set. We first consider the problem of finding a single plane Hamiltonian path pi with endpoints s, t in S and constraints given by a segment ab, where a, b in S. We consider the following scenarios: (i) ab in pi; (ii) ab not in pi. We characterize those quintuples (S, a, b, s, t) for which pi exists. Secondly, we consider the problem of finding two plane Hamiltonian paths pi_1, pi_2 on a set S with constraints given by a segment ab, where a, b in S. We consider the following scenarios: (i) pi_1 and pi_2 share no edges and ab is an edge of pi_1; (ii) pi_1 and pi_2 share no edges and none of them includes ab as an edge; (iii) both pi_1 and pi_2 include ab as an edge and share no other edges. In all cases, we characterize those triples (S, a, b) for which pi_1 and pi_2 exist.

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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

Computational complexity of covering regular trees

A graph covering projection, also referred to as a locally bijective homomorphism, is a mapping between the vertices and edges of two graphs that preserves incidences and is a local bijection. This concept originates in topological graph theory but has also found applications in combinatorics and theoretical computer science. In this paper we consider undirected graphs in the most general setting -- graphs may contain multiple edges, loops, and semi-edges. This is in line with recent trends in topological graph theory and mathematical physics. We advance the study of the computational complexity of the {\sc $H$-Cover} problem, which asks whether an input graph allows a covering projection onto a parameter graph $H$. The quest for a complete characterization started in 1990's. Several results for simple graphs or graphs without semi-edges have been known, the role of semi-edges in the complexity setting has started to be investigated only recently. One of the most general known NP-hardness results states that {\sc $H$}-Cover is NP-complete for every simple connected regular graph of valency greater than two. We complement this result by considering regular graphs $H$ arising from connected acyclic graphs by adding semi-edges. Namely, we prove that any graph obtained by adding semi-edges to the vertices of a tree making it a $d$-regular graph with $d \geq 3$, defines an NP-complete graph covering problem. In line with the so called Strong Dichotomy Conjecture, we prove that the NP-hardness holds even for simple graphs on input.

cs.DM↗

arXiv2025

Three Edge-disjoint Plane Spanning Paths in a Point Set

We consider the following problem: Given a set $S$ of $n$ distinct points in the plane, how many edge-disjoint plane straight-line spanning paths can be drawn on $S$? Each spanning path must be crossing-free, but edges from different paths are allowed to intersect at arbitrary points. It is known that if the points of $S$ are in convex position, then $\lfloor n/2 \rfloor$ such paths always exist. However, for general point sets, the best known construction yields only two edge-disjoint plane spanning paths. In this paper, we prove that for any set $S$ of at least ten points in general position (i.e., no three points are collinear), it is always possible to draw at least three edge-disjoint plane straight-line spanning paths. Our proof relies on a structural result about halving lines in point sets and builds on the known two-path construction, which we also strengthen: we show that for any set $S$ of at least six points, and for any two specified points on the boundary of the convex hull of $S$, there exist two edge-disjoint plane spanning paths that start at those prescribed points. Finally, we complement our positive results with a lower bound: for every $n \geq 6$, there exists a set of $n$ points for which no more than $\lceil n/3 \rceil$ edge-disjoint plane spanning paths are possible.

cs.CG↗

arXiv2025

Hamiltonian path and Hamiltonian cycle are solvable in polynomial time in graphs of bounded independence number

A Hamiltonian path (a Hamiltonian cycle) in a graph is a path (a cycle, respectively) that traverses all of its vertices. The problems of deciding their existence in an input graph are well-known to be NP-complete, in fact, they belong to the first problems shown to be computationally hard when the theory of NP-completeness was being developed. A lot of research has been devoted to the complexity of Hamiltonian path and Hamiltonian cycle problems for special graph classes, yet only a handful of positive results are known. The complexities of both of these problems have been open even for $4K_1$-free graphs, i.e., graphs of independence number at most $3$. We answer this question in the general setting of graphs of bounded independence number. We also consider a newly introduced problem called \emph{Hamiltonian-$\ell$-Linkage} which is related to the notions of a path cover and of a linkage in a graph. This problem asks if given $\ell$ pairs of vertices in an input graph can be connected by disjoint paths that altogether traverse all vertices of the graph. For $\ell=1$, Hamiltonian-1-Linkage asks for existence of a Hamiltonian path connecting a given pair of vertices. Our main result reads that for every pair of integers $k$ and $\ell$, the Hamiltonian-$\ell$-Linkage problem is polynomial time solvable for graphs of independence number not exceeding $k$.

cs.DM↗

arXiv2025

Computational Complexity of Covering Colored Mixed Multigraphs with Simple Degree Partitions

The notion of graph covers (also referred to as locally bijective homomorphisms) plays an important role in topological graph theory and has found its computer science applications in models of local computation. For a fixed target graph $H$, the {\sc $H$-Cover} problem asks if an input graph $G$ allows a graph covering projection onto $H$. Despite the fact that the quest for characterizing the computational complexity of {\sc $H$-Cover} had been started more than 30 years ago, only a handful of general results have been known so far. In this paper, we present a complete characterization of the computational complexity of covering coloured graphs for the case that every equivalence class in the degree partition of the target graph has at most two vertices. We prove this result in a very general form. Following the lines of current development of topological graph theory, we study graphs in the most relaxed sense of the definition. In particular, we consider graphs that are mixed (they may have both directed and undirected edges), may have multiple edges, loops, and semi-edges. We show that a strong P/NP-complete dichotomy holds true in the sense that for each such fixed target graph $H$, the {\sc $H$-Cover} problem is either polynomial-time solvable for arbitrary inputs, or NP-complete even for simple input graphs.

cs.DM↗

arXiv2024

The Parametrized Complexity of the Segment Number

Given a straight-line drawing of a graph, a segment is a maximal set of edges that form a line segment. Given a planar graph $G$, the segment number of $G$ is the minimum number of segments that can be achieved by any planar straight-line drawing of $G$. The line cover number of $G$ is the minimum number of lines that support all the edges of a planar straight-line drawing of $G$. Computing the segment number or the line cover number of a planar graph is $\exists\mathbb{R}$-complete and, thus, NP-hard. We study the problem of computing the segment number from the perspective of parameterized complexity. We show that this problem is fixed-parameter tractable with respect to each of the following parameters: the vertex cover number, the segment number, and the line cover number. We also consider colored versions of the segment and the line cover number.

cs.CG↗

arXiv2024

On the Structure of Hamiltonian Graphs with Small Independence Number

A Hamiltonian path (cycle) in a graph is a path (cycle, respectively) which passes through all of its vertices. The problems of deciding the existence of a Hamiltonian cycle (path) in an input graph are well known to be NP-complete, and restricted classes of graphs which allow for their polynomial-time solutions are intensively investigated. Until very recently the complexity was open even for graphs of independence number at most 3. So far unpublished result of Jedličková and Kratochv\'ıl [arXiv:2309.09228] shows that for every integer $k$, Hamiltonian path and cycle are polynomial-time solvable in graphs of independence number bounded by $k$. As a companion structural result, we determine explicit obstacles for the existence of a Hamiltonian path for small values of $k$, namely for graphs of independence number 2, 3, and 4. Identifying these obstacles in an input graph yields alternative polynomial-time algorithms for Hamiltonian path and cycle with no large hidden multiplicative constants.

math.CO↗

arXiv2024

On a Combinatorial Problem Arising in Machine Teaching

We study a model of machine teaching where the teacher mapping is constructed from a size function on both concepts and examples. The main question in machine teaching is the minimum number of examples needed for any concept, the so-called teaching dimension. A recent paper [7] conjectured that the worst case for this model, as a function of the size of the concept class, occurs when the consistency matrix contains the binary representations of numbers from zero and up. In this paper we prove their conjecture. The result can be seen as a generalization of a theorem resolving the edge isoperimetry problem for hypercubes [12], and our proof is based on a lemma of [10].

math.CO↗

arXiv2023

Computational Complexity of Covering Disconnected Multigraphs

The notion of graph covers is a discretization of covering spaces introduced and deeply studied in topology. In discrete mathematics and theoretical computer science, they have attained a lot of attention from both the structural and complexity perspectives. Nonetheless, disconnected graphs were usually omitted from the considerations with the explanation that it is sufficient to understand coverings of the connected components of the target graph by components of the source one. However, different (but equivalent) versions of the definition of covers of connected graphs generalize to non-equivalent definitions for disconnected graphs. The aim of this paper is to summarize this issue and to compare three different approaches to covers of disconnected graphs: 1) locally bijective homomorphisms, 2) globally surjective locally bijective homomorphisms (which we call \emph{surjective covers}), and 3) locally bijective homomorphisms which cover every vertex the same number of times (which we call \emph{equitable covers}). The standpoint of our comparison is the complexity of deciding if an input graph covers a fixed target graph. We show that both surjective and equitable covers satisfy what certainly is a natural and welcome property: covering a disconnected graph is polynomial-time decidable if such it is for every connected component of the graph, and it is NP-complete if it is NP-complete for at least one of its components. We further argue that the third variant, equitable covers, is the most natural one, namely when considering covers of colored graphs. Moreover, the complexity of surjective and equitable covers differ from the fixed parameter complexity point of view. In line with the current trends in topological graph theory, as well as its applications in mathematical physics, we consider graphs in a very general sense[...]

cs.DM↗

arXiv2023

List covering of regular multigraphs with semi-edges

In line with the recent development in topological graph theory, we are considering undirected graphs that are allowed to contain {\em multiple edges}, {\em loops}, and {\em semi-edges}. A graph is called {\em simple} if it contains no semi-edges, no loops, and no multiple edges. A graph covering projection, also known as a locally bijective homomorphism, is a mapping between vertices and edges of two graphs which preserves incidences and which is a local bijection on the edge-neighborhood of every vertex. This notion stems from topological graph theory, but has also found applications in combinatorics and theoretical computer science. It has been known that for every fixed simple regular graph $H$ of valency greater than 2, deciding if an input graph covers $H$ is NP-complete. Graphs with semi-edges have been considered in this context only recently and only partial results on the complexity of covering such graphs are known so far. In this paper we consider the list version of the problem, called \textsc{List-$H$-Cover}, where the vertices and edges of the input graph come with lists of admissible targets. Our main result reads that the \textsc{List-$H$-Cover} problem is NP-complete for every regular graph $H$ of valency greater than 2 which contains at least one semi-simple vertex (i.e., a vertex which is incident with no loops, with no multiple edges and with at most one semi-edge). Using this result we show the NP-co/polytime dichotomy for the computational complexity of \textsc{ List-$H$-Cover} for cubic graphs.

cs.DM↗

arXiv2022

Beyond circular-arc graphs -- recognizing lollipop graphs and medusa graphs

In 1992 Biró, Hujter and Tuza introduced, for every fixed connected graph $H$, the class of $H$-graphs, defined as the intersection graphs of connected subgraphs of some subdivision of $H$. Recently, quite a lot of research has been devoted to understanding the tractability border for various computational problems, such as recognition or isomorphism testing, in classes of $H$-graphs for different graphs $H$. In this work we undertake this research topic, focusing on the recognition problem. Chaplick, Töpfer, Voborn\'ık, and Zeman showed, for every fixed tree $T$, a polynomial-time algorithm recognizing $T$-graphs. Tucker showed a polynomial time algorithm recognizing $K_3$-graphs (circular-arc graphs). On the other hand, Chaplick at al. showed that recognition of $H$-graphs is $NP$-hard if $H$ contains two different cycles sharing an edge. The main two results of this work narrow the gap between the $NP$-hard and $P$ cases of $H$-graphs recognition. First, we show that recognition of $H$-graphs is $NP$-hard when $H$ contains two different cycles. On the other hand, we show a polynomial-time algorithm recognizing $L$-graphs, where $L$ is a graph containing a cycle and an edge attached to it ($L$-graphs are called lollipop graphs). Our work leaves open the recognition problems of $M$-graphs for every unicyclic graph $M$ different from a cycle and a lollipop. Other results of this work, which shed some light on the cases that remain open, are as follows. Firstly, the recognition of $M$-graphs, where $M$ is a fixed unicyclic graph, admits a polynomial time algorithm if we restrict the input to graphs containing particular holes (hence recognition of $M$-graphs is probably most difficult for chordal graphs). Secondly, the recognition of medusa graphs, which are defined as the union of $M$-graphs, where $M$ runs over all unicyclic graphs, is $NP$-complete.

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arXiv2020

U-Bubble Model for Mixed Unit Interval Graphs and its Applications: The MaxCut Problem Revisited

Interval graphs, intersection graphs of segments on a real line (intervals), play a key role in the study of algorithms and special structural properties. Unit interval graphs, their proper subclass, where each interval has a unit length, has also been extensively studied. We study mixed unit interval graphs---a generalization of unit interval graphs where each interval has still a unit length, but intervals of more than one type (open, closed, semi-closed) are allowed. This small modification captures a much richer class of graphs. In particular, mixed unit interval graphs are not claw-free, compared to unit interval graphs. Heggernes, Meister, and Papadopoulos defined a representation of unit interval graphs called the bubble model which turned out to be useful in algorithm design. We extend this model to the class of mixed unit interval graphs and demonstrate the advantages of this generalized model by providing a subexponential-time algorithm for solving the MaxCut problem on mixed unit interval graphs. In addition, we derive a polynomial-time algorithm for certain subclasses of mixed unit interval graphs. We point out a substantial mistake in the proof of the polynomiality of the MaxCut problem on unit interval graphs by Boyaci, Ekim, and Shalom (2017). Hence, the time complexity of this problem on unit interval graphs remains open. We further provide a better algorithmic upper-bound on the clique-width of mixed unit interval graphs. Clique-width is one of the most general structural graph parameters, where a large group of natural problems is still solvable in the tractable time when an efficient representation is given. Unfortunately, the exact computation of the clique-width representation is \NP-hard. Therefore, good upper-bounds on clique-width are highly appreciated, in particular, when such a bound is algorithmic.

cs.DS↗

arXiv2018

3-connected Reduction for Regular Graph Covers

A graph $G$ covers a graph $H$ if there exists a locally bijective homomorphism from $G$ to $H$. We deal with regular coverings in which this homomorphism is prescribed by an action of a semiregular subgroup $Γ$ of $\textrm{Aut}(G)$; so $H \cong G / Γ$. In this paper, we study the behaviour of regular graph covering with respect to 1-cuts and 2-cuts in $G$. We describe reductions which produce a series of graphs $G = G_0,\dots,G_r$ such that $G_{i+1}$ is created from $G_i$ by replacing certain inclusion minimal subgraphs with colored edges. The process ends with a primitive graph $G_r$ which is either 3-connected, or a cycle, or $K_2$. This reduction can be viewed as a non-trivial modification of reductions of Mac Lane (1937), Trachtenbrot (1958), Tutte (1966), Hopcroft and Tarjan (1973), Cuningham and Edmonds (1980), Walsh (1982), and others. A novel feature of our approach is that in each step all essential information about symmetries of $G$ are preserved. A regular covering projection $G_0\to H_0$ induces regular covering projections $G_i \to H_i$ where $H_i$ is the $i$-th quotient reduction of $H_0$. This property allows to construct all possible quotients $H_0$ of $G_0$ from the possible quotients $H_r$ of $G_r$. By applying this method to planar graphs, we give a proof of Negami's Theorem (1988). Our structural results are also used in subsequent papers for regular covering testing when $G$ is a planar graph and for an inductive characterization of the automorphism groups of planar graphs (see Babai (1973) as well).

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