arXiv subjects

Alexander Firbas

Publications and source records attributed to Alexander Firbas.

8 recordsLinked to original sources

arXiv2026

The (Parameterized) Complexity of Ordering a Graph While Avoiding a Forbidden Pattern

In this paper, we study the Pattern Avoidance problem of determining whether a given graph $G$ admits a linear vertex order which avoids a given pattern $P$, i.e., a vertex sequence with some forced and forbidden edges, on every suborder. Such patterns form a natural ordered counterpart to induced subgraphs in the order-invariant setting, and it is known that Pattern Avoidance captures a broad variety of graph problems including Bandwidth, Vertex Coloring, Queue Number, and extends to vertex-deletion problems such as Odd Cycle Transversal. We show that Pattern Avoidance is $\Sigma_2^{\textsf{P}}$-complete and furthermore remains intractable (in both the classical and parameterized sense) even under a variety of severe restrictions to both the pattern $P$ and the graph $G$. As our main contributions, we complement these lower bounds with the following tractability results, which provide a unifying framework for recognizing pattern-definable graph classes: - a fixed-parameter algorithm w.r.t. the vertex integrity of $G$ plus $|V(P)|$, - a fixed-parameter algorithm w.r.t. the neighborhood diversity of $G$ plus $|E(P)|$, and - a polynomial algorithm for Pattern Avoidance on forests for almost all constant-sized patterns.

cs.DS↗

arXiv2026

Not All Degree Constraints Are Created Equal when Computing Spanning Trees

We study the computation of minimum spanning trees subject to local degree constraints. Recent work (ICALP 2026) established that three natural formalizations of this problem share the exact same parameterized complexity under standard structural graph parameters, including treewidth, pathwidth and clique-width. This applies to the cases where every vertex has a single target degree (Specified Degree MST), or a degree upper bound (Bounded Degree MST), or is equipped with a set of admissible degrees (Set of Degrees MST). In this paper, we investigate these problems under more restrictive parameterizations and reveal that their complexity landscapes fundamentally diverge on bounded-treedepth graphs. Specifically, we prove that the former two problems are fixed-parameter tractable when parameterized by the treedepth of the input graph. In sharp contrast, we show that Set of Degrees MST remains W[1]-hard parameterized by treedepth, even when combined with the feedback vertex number (i.e., deletion distance to treewidth $1$). Finally, we show that this divergence seems to be specific to treedepth: we exclude an analogous W[1]-hardness result for Set of Degrees MST w.r.t. the vertex cover number and also rule out fixed-parameter algorithms for the former two problems w.r.t. deletion distance to constant pathwidth.

cs.DS↗

arXiv2026

Two-Layer Drawings with a Tree on Top: Vertex Splits and Fixed-Parameter Algorithms

Two-layer drawings of bipartite graphs place the vertices of each part on one of two parallel lines and draw the edges as straight-line links. Traditionally, the optimization goal is to find vertex permutations on one or both layers that minimize the induced number of edge crossings. This problem is NP-hard, and crossing-minimal solutions may still contain many crossings. Recently, there has been growing interest in an orthogonal optimization goal, namely removing all crossings by vertex splitting, i.e., replacing original vertices by two or more copies and distributing the adjacencies among them. In this paper, we study a natural extension of the two-layer vertex splitting problem in which the vertex order on one layer is constrained by a given auxiliary tree $T$, motivated by applications such as the visualization of anatomical hierarchies in the Human Reference Atlas. We investigate the parameterized complexity of this problem and obtain two main contributions: (1) a fixed-parameter algorithm with respect to the number $k$ of splits, and (2) an ETH-tight single-exponential fixed-parameter algorithm with respect to the maximum degree of $T$. Moreover, we build on the latter result to obtain an ETH-tight single-exponential algorithm for the classical unconstrained version of the problem, improving upon the previous $O^*(2^{k\cdot \log k})$ algorithms. Finally, we also implement our algorithm and show that it performs well in practice.

cs.CG↗

arXiv2026

The Parameterized Complexity of Geometric 1-Planarity

A graph is geometric 1-planar if it admits a straight-line drawing where each edge is crossed at most once. We provide the first systematic study of the parameterized complexity of recognizing geometric 1-planar graphs. By substantially extending a technique of Bannister, Cabello, and Eppstein, combined with Thomassen's characterization of 1-planar embeddings that can be straightened, we show that the problem is fixed-parameter tractable when parameterized by treedepth. Furthermore, we obtain a kernel for Geometric 1-Planarity parameterized by the feedback edge number $\ell$. As a by-product, we improve the best known kernel size of $O((3\ell)!)$ for 1-Planarity and $k$-Planarity under the same parameterization to $O(\ell \cdot 8^{\ell})$. Our approach naturally extends to Geometric $k$-Planarity, yielding a kernelization under the same parameterization, albeit with a larger kernel. Complementing these results, we provide matching lower bounds: Geometric 1-Planarity remains \NP-complete even for graphs of bounded pathwidth, bounded feedback vertex number, and bounded bandwidth.

cs.CC↗

arXiv2025

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↗

arXiv2024

Pathways to Tractability for Geometric Thickness

We study the classical problem of computing geometric thickness, i.e., finding a straight-line drawing of an input graph and a partition of its edges into as few parts as possible so that each part is crossing-free. Since the problem is NP-hard, we investigate its tractability through the lens of parameterized complexity. As our first set of contributions, we provide two fixed-parameter algorithms which utilize well-studied parameters of the input graph, notably the vertex cover and feedback edge numbers. Since parameterizing by the thickness itself does not yield tractability and the use of other structural parameters remains open due to general challenges identified in previous works, as our second set of contributions, we propose a different pathway to tractability for the problem: extension of partial solutions. In particular, we establish a full characterization of the problem's parameterized complexity in the extension setting depending on whether we parameterize by the number of missing vertices, edges, or both.

cs.CC↗

arXiv2024

On the Complexity of Establishing Hereditary Graph Properties via Vertex Splitting

Vertex splitting is a graph operation that replaces a vertex $v$ with two nonadjacent new vertices and makes each neighbor of $v$ adjacent with one or both of the introduced vertices. Vertex splitting has been used in contexts from circuit design to statistical analysis. In this work, we explore the computational complexity of achieving a given graph property $\Pi$ by a limited number of vertex splits, formalized as the problem $\Pi$ Vertex Splitting ($\Pi$-VS). We focus on hereditary graph properties and contribute four groups of results: First, we classify the classical complexity of $\Pi$-VS for graph properties characterized by forbidden subgraphs of size at most 3. Second, we provide a framework that allows to show NP-completeness whenever one can construct a combination of a forbidden subgraph and prescribed vertex splits that satisfy certain conditions. Leveraging this framework we show NP-completeness when $\Pi$ is characterized by forbidden subgraphs that are sufficiently well connected. In particular, we show that $F$-Free-VS is NP-complete for each biconnected graph $F$. Third, we study infinite families of forbidden subgraphs, obtaining NP-hardness for Bipartite-VS and Perfect-VS. Finally, we touch upon the parameterized complexity of $\Pi$-VS with respect to the number of allowed splits, showing para-NP-hardness for $K_3$-Free-VS and deriving an XP-algorithm when each vertex is only allowed to be split at most once.

cs.CC↗

arXiv2023

The Complexity of Cluster Vertex Splitting and Company

Clustering a graph when the clusters can overlap can be seen from three different angles: We may look for cliques that cover the edges of the graph with bounded overlap, we may look to add or delete few edges to uncover the cluster structure, or we may split vertices to separate the clusters from each other. Splitting a vertex $v$ means to remove it and to add two new copies of $v$ and to make each previous neighbor of $v$ adjacent with at least one of the copies. In this work, we study underlying computational problems regarding the three angles to overlapping clusterings, in particular when the overlap is small. We show that the above-mentioned covering problem is NP-complete. We then make structural observations that show that the covering viewpoint and the vertex-splitting viewpoint are equivalent, yielding NP-hardness for the vertex-splitting problem. On the positive side, we show that splitting at most $k$ vertices to obtain a cluster graph has a problem kernel with $O(k)$ vertices. Finally, we observe that combining our hardness results with the so-called critical-clique lemma yields NP-hardness for Cluster Editing with Vertex Splitting, which was previously open (Abu-Khzam et al. [ISCO 2018]) and independently shown to be NP-hard by Arrighi et al. [IPEC 2023]. We observe that a previous version of the critical-clique lemma was flawed; a corrected version has appeared in the meantime on which our hardness result is based.

cs.DS↗