arXiv subjects

Krisztina Szilágyi

Publications and source records attributed to Krisztina Szilágyi.

10 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.

cs.CG↗

arXiv2026

Fine-Grained Complexity of Computing Degree-Constrained Spanning Trees

We investigate the computation of minimum-cost spanning trees satisfying prescribed vertex degree constraints: Given a graph $G$ and a constraint function $D$, we ask for a (minimum-cost) spanning tree $T$ such that for each vertex $v$, $T$ achieves a degree specified by $D(v)$. Specifically, we consider three kinds of constraint functions ordered by their generality -- $D$ may either assign each vertex to a list of admissible degrees, an upper bound on the degrees, or a specific degree. Using a combination of novel techniques and state-of-the-art machinery, we obtain an almost-complete overview of the fine-grained complexity of these problems taking into account the most classical graph parameters of the input graph $G$. In particular, we present SETH-tight upper and lower bounds for these problems when parameterized by the pathwidth and cutwidth, an ETH-tight algorithm parameterized by the cliquewidth, and a nearly SETH-tight algorithm parameterized by treewidth. In order to obtain our upper bound for clique-width, we develop a novel technique of double representation through ``requirement shifting''. Using this technique, we also obtain an ETH-tight single-exponential XP algorithm for the Exact Leaf Spanning Tree problem parameterized by clique-width, which settles the final remaining open case for clique-width from the classical Cut and Count of Cygan et al. [FOCS 2011, TALG 2022]. This shows the versatility of our technique and its potential applicability to other problems as well. Additionally, in order to establish our lower and upper bounds we introduce a number of tools which may be of independent interest, including lazy coloring and ``asymptotic'' SETH-based reductions for structural parameters.

cs.DS↗

arXiv2026

A Polynomial Kernel for Face Cover on Non-Embedded Planar Graphs

Given a planar graph, a subset of its vertices called terminals, and $k \in \mathbb{N}$, the Face Cover Number problem asks whether the terminals lie on the boundaries of at most $k$ faces of some embedding of the input graph. When a plane graph is given in the input, the problem is known to have a polynomial kernel~\cite{GarneroST17}. In this paper, we present the first polynomial kernel for Face Cover Number when the input is a planar graph (without a fixed embedding). Our approach overcomes the challenge of not having a predefined set of face boundaries by building a kernel bottom-up on an SPR-tree while preserving the essential properties of the face cover along the way.

cs.DS↗

arXiv2025

Pathfinding in Self-Deleting Graphs

In this paper, we study the problem of pathfinding on traversal-dependent graphs, i.e., graphs whose edges change depending on the previously visited vertices. In particular, we study \emph{self-deleting graphs}, introduced by Carmesin et al. (Sarah Carmesin, David Woller, David Parker, Miroslav Kulich, and Masoumeh Mansouri. The Hamiltonian cycle and travelling salesperson problems with traversal-dependent edge deletion. J. Comput. Sci.), which consist of a graph $G=(V, E)$ and a function $f\colon V\rightarrow 2^E$, where $f(v)$ is the set of edges that will be deleted after visiting the vertex $v$. In the \textsc{(Shortest) Self-Deleting $s$-$t$-path} problem we are given a self-deleting graph and its vertices $s$ and $t$, and we are asked to find a (shortest) path from $s$ to $t$, such that it does not traverse an edge in $f(v)$ after visiting $v$ for any vertex $v$. We prove that \textsc{Self-Deleting $s$-$t$-path} is NP-hard even if the given graph is outerplanar, bipartite, has maximum degree $3$, bandwidth $2$ and $|f(v)|\leq 1$ for each vertex $v$. We show that \textsc{Shortest Self-Deleting $s$-$t$-path} is W[1]-complete parameterized by the length of the sought path and that \textsc{Self-Deleting $s$-$t$-path} is \W{1}-complete parameterized by the vertex cover number, feedback vertex set number and treedepth. We also show that the problem becomes FPT when we parameterize by the maximum size of $f(v)$ and several structural parameters. Lastly, we show that the problem does not admit a polynomial kernel even for parameterization by the vertex cover number and the maximum size of $f(v)$ combined already on 2-outerplanar graphs.

cs.DS↗

arXiv2025

Density of Traceable Graphs

We establish tight lower and upper bounds on the number of edges in traceable graphs in several classes of dense graphs. A graph is traceable if it has a Hamiltonian path. We show that the bound is: - quadratic for the class of graphs of bounded neighborhood diversity, bounded size of maximum induced matching or bounded cluster vertex deletion number; - n log n for the class of cographs or, more generaly, bounded modular-width, and for the class of bounded distance to cograph; and - sligthly superlinear for the class of bounded shrub-depth.

math.CO↗

arXiv2025

XALP-completeness of Parameterized Problems on Planar Graphs

The class XNLP consists of (parameterized) problems that can be solved nondeterministically in $f(k)n^{O(1)}$ time and $f(k)\log n$ space, where $n$ is the size of the input instance and $k$ the parameter. The class XALP consists of problems that can be solved in the above time and space with access to an additional stack. These two classes are a "natural home" for many standard graph problems and their generalizations. In this paper, we show the hardness of several problems on planar graphs, parameterized by outerplanarity, treewidth and pathwidth, thus strengthening several existing results. In particular, we show the XALP-completeness of the following problems parameterized by outerplanarity: All-or-Nothing Flow, Target Outdegree Orientation, Capacitated (Red-Blue) Dominating Set, Target Set Selections etc. We also show the XNLP-completeness of Scattered Set parameterized by pathwidth and XALP-completeness parameterized by treewidth and outerplanarity.

cs.CC↗

arXiv2024

Algorithms and Turing Kernels for Detecting and Counting Small Patterns in Unit Disk Graphs

In this paper we investigate the parameterized complexity of the task of counting and detecting occurrences of small patterns in unit disk graphs: Given an $n$-vertex unit disk graph $G$ with an embedding of ply $p$ (that is, the graph is represented as intersection graph with closed disks of unit size, and each point is contained in at most $p$ disks) and a $k$-vertex unit disk graph $P$, count the number of (induced) copies of $P$ in $G$. For general patterns $P$, we give an $2^{O(p k /\log k)}n^{O(1)}$ time algorithm for counting pattern occurrences. We show this is tight, even for ply $p=2$ and $k=n$: any $2^{o(n/\log n)}n^{O(1)}$ time algorithm violates the Exponential Time Hypothesis (ETH). For most natural classes of patterns, such as connected graphs and independent sets we present the following results: First, we give an $(pk)^{O(\sqrt{pk})}n^{O(1)}$ time algorithm, which is nearly tight under the ETH for bounded ply and many patterns. Second, for $p= k^{O(1)}$ we provide a Turing kernelization (i.e. we give a polynomial time preprocessing algorithm to reduce the instance size to $k^{O(1)}$). Our approach combines previous tools developed for planar subgraph isomorphism such as `efficient inclusion-exclusion' from [Nederlof STOC'20], and `isomorphisms checks' from [Bodlaender et al. ICALP'16] with a different separator hierarchy and a new bound on the number of non-isomorphic separations of small order tailored for unit disk graphs.

cs.CC↗

arXiv2023

Parameterized Algorithms for Covering by Arithmetic Progressions

An arithmetic progression is a sequence of integers in which the difference between any two consecutive elements is the same. We investigate the parameterized complexity of two problems related to arithmetic progressions, called Cover by Arithmetic Progressions (CAP) and Exact Cover by Arithmetic Progressions (XCAP). In both problems, we are given a set $X$ consisting of $n$ integers along with an integer $k$, and our goal is to find $k$ arithmetic progressions whose union is $X$. In XCAP we additionally require the arithmetic progressions to be disjoint. Both problems were shown to be NP-complete by Heath [IPL'90]. We present a $2^{O(k^2)} poly(n)$ time algorithm for CAP and a $2^{O(k^3)} poly(n)$ time algorithm for XCAP. We also give a fixed parameter tractable algorithm for CAP parameterized below some guaranteed solution size. We complement these findings by proving that CAP is Strongly NP-complete in the field $\mathbb{Z}_p$, if $p$ is a prime number part of the input.

cs.CC↗

arXiv2023

Tight bounds for counting colorings and connected edge sets parameterized by cutwidth

We study the fine-grained complexity of counting the number of colorings and connected spanning edge sets parameterized by the cutwidth and treewidth of the graph. While decompositions of small treewidth decompose the graph with small vertex separators, decompositions with small cutwidth decompose the graph with small \emph{edge} separators. Let $p,q \in \mathbb{N}$ such that $p$ is a prime and $q \geq 3$. - If $p$ divides $q-1$, there is a $(q-1)^{\text{ctw}}n^{O(1)}$ time algorithm for counting list $q$-colorings modulo $p$ of $n$-vertex graphs of cutwidth $\text{ctw}$ and for all $\varepsilon>0$ there is no algorithm running in time $(q-1-\varepsilon)^{\text{ctw}} n^{O(1)}$, assuming the Strong Exponential Time Hypothesis (SETH). - If $p$ does not divide $q-1$, there is a (folklore) $q^{\text{ctw}}n^{O(1)}$ time algorithm for counting list $q$-colorings modulo $p$ of $n$-vertex graphs of cutwidth $\text{ctw}$ and for all $\varepsilon>0$ there is no algorithm running in time $(q-\varepsilon)^{\text{ctw}} n^{O(1)}$, assuming SETH. The lower bounds are in stark contrast with the existing $2^{\text{ctw}}n^{O(1)}$ time algorithm to compute the chromatic number of a graph by Jansen and Nederlof~[Theor. Comput. Sci.'18]. Both our algorithms and lower bounds employ use of the matrix rank method, by relating the complexity of the problem to the rank of a certain `compatibility matrix' in a non-trivial way. We extend our lower bounds to counting connected spanning edge sets modulo $p$ and give an algorithm with matching running time for both treewidth and cutwidth.

cs.DS↗

arXiv2021

On the Parameterized Complexity of the Connected Flow and Many Visits TSP Problem

We study a variant of Min Cost Flow in which the flow needs to be connected. Specifically, in the Connected Flow problem one is given a directed graph $G$, along with a set of demand vertices $D \subseteq V(G)$ with demands $\mathsf{dem}: D \rightarrow \mathbb{N}$, and costs and capacities for each edge. The goal is to find a minimum cost flow that satisfies the demands, respects the capacities and induces a (strongly) connected subgraph. This generalizes previously studied problems like the (Many Visits) TSP. We study the parameterized complexity of Connected Flow parameterized by $|D|$, the treewidth $tw$ and by vertex cover size $k$ of $G$ and provide: (i) $\mathsf{NP}$-completeness already for the case $|D|=2$ with only unit demands and capacities and no edge costs, and fixed-parameter tractability if there are no capacities, (ii) a fixed-parameter tractable $\mathcal{O}^{\star}(k^{\mathcal{O}(k)})$ time algorithm for the general case, and a kernel of size polynomial in $k$ for the special case of Many Visits TSP, (iii) a $|V(G)|^{\mathcal{O}(tw)}$ time algorithm and a matching $|V(G)|^{o(tw)}$ time conditional lower bound conditioned on the Exponential Time Hypothesis. To achieve some of our results, we significantly extend an approach by Kowalik et al.~[ESA'20].

cs.DS↗