arXiv subjects

Yuto Okada

Publications and source records attributed to Yuto Okada.

15 recordsLinked to original sources

arXiv2026

Treewidth of the $n \times n$ toroidal grid

In this paper, we show that the treewidth of the $n \times n$ toroidal grid is $2n-1$ for all $n \ge 5$. This closes the gap between the previously known upper bound of $2n-1$ (Ellis and Warren, DAM 2008) and the lower bound of $2n-2$ (Kiyomi, Okamoto, and Otachi, DAM 2016). To establish the matching lower bound, we construct a bramble of maximum order by utilizing maximum components obtained after removing $2n-1$ vertices. Our construction relies on the vertex-isoperimetric properties of the infinite grid to establish tight lower bounds on neighborhood sizes, combined with a careful analysis of balls of radius $n/2-1$ and their boundaries to overcome structural obstructions when $n$ is even.

math.CO↗

arXiv2026

Min-1-Planarity is NP-Hard

In this paper, we show that it is NP-hard to determine whether a given graph admits a min-1-planar drawing. A drawing of a graph is min-$k$-planar if, for every crossing in the drawing, at least one of the two crossing edges involves at most $k$ crossings. This notion of min-$k$-planarity was introduced by Binucci, B\"{u}ngener, Di Battista, Didimo, Dujmovi\'c, Hong, Kaufmann, Liotta, Morin, and Tappini [GD 2023; JGAA, 2024] as a generalization of $k$-planarity.

cs.CG↗

arXiv2026

On Edge-Disjoint Maximal Outerplanar Graphs

We provide two constructions for $t$ edge-disjoint maximal outerplanar graphs on every number of $n \geq 4t$ vertices. The bound on the minimum number of vertices is tight. These constructions yield the existence of optimal outerthickness-$t$ graphs for every $t \in \mathbb{N}$. While one of the constructions works for all values of $t$ and extends graphs from Guy and Nowakowski (1990), the other one holds only for powers of $2$, but yields graphs with maximum degree logarithmic in the number of vertices. Thus, the latter may be helpful in tackling the open question of determining the outerthickness of all complete graphs.

math.CO↗

arXiv2025

Finding a Maximum Common (Induced) Subgraph: Structural Parameters Revisited

We study the parameterized complexity of the problems of finding a maximum common (induced) subgraph of two given graphs. Since these problems generalize several NP-complete problems, they are intractable even when parameterized by strongly restricted structural parameters. Our contribution in this paper is to sharply complement the hardness of the problems by showing fixed-parameter tractable cases: both induced and non-induced problems parameterized by max-leaf number and by neighborhood diversity, and the induced problem parameterized by twin cover number. These results almost completely determine the complexity of the problems with respect to well-studied structural parameters. Also, the result on the twin cover number presents a rather rare example where the induced and non-induced cases have different complexity.

cs.DS↗

arXiv2025

One-Sided Local Crossing Minimization

Drawing graphs with the minimum number of crossings is a classical problem that has been studied extensively. Many restricted versions of the problem have been considered. For example, bipartite graphs can be drawn such that the two sets in the bipartition of the vertex set are mapped to two parallel lines, and the edges are drawn as straight-line segments. In this setting, the number of crossings depends only on the ordering of the vertices on the two lines. Two natural variants of the problem have been studied. In the one-sided case, the order of the vertices on one of the two lines is given and fixed; in the two-sided case, no order is given. Both cases are important yet NP-hard subproblems in the so-called Sugiyama framework for drawing layered graphs with few crossings. For the one-sided case, Eades and Wormald [Algorithmica 1994] introduced a median heuristic and showed that it has an approximation ratio of 3. In recent years, researchers have focused on a local version of crossing minimization, where the aim is to minimize the maximum number of crossings per edge instead of the total number of crossings. Kobayashi, Okada, and Wolff [SoCG 2025] investigated the complexity of local crossing minimization parameterized by the natural parameter. They conjectured that one-sided local crossing minimization is NP-hard. In this work, we confirm their conjecture by showing that the problem is NP-hard even for forests of high-degree stars. In fact, more strongly, the reduction yields a tight lower bound, which excludes the existence of subexponential-time algorithms assuming the Exponential-Time Hypothesis. In contrast, we present a quadratic-time algorithm for the special case of forests of stars of maximum degree 2. Finally, we provide a median heuristic with a carefully designed tie-breaking scheme and prove that it has an approximation ratio of 3 in the local setting.

cs.DS↗

arXiv2025

Forcing a unique minimum spanning tree and a unique shortest path

A forcing set $S$ in a combinatorial problem is a set of elements such that there is a unique solution that contains all the elements in $S$. An anti-forcing set is the symmetric concept: a set $S$ of elements is called an anti-forcing set if there is a unique solution disjoint from $S$. There are extensive studies on the computational complexity of finding a minimum forcing set in various combinatorial problems, and the known results indicate that many problems are harder than their classical counterparts: the decision version of finding a minimum forcing set for perfect matchings is NP-complete [Adams et al., Discrete Mathematics, 2004], and that of finding a minimum forcing set for satisfying assignments for 3CNF formulas is $\Sigma_2^P$-complete [Hatami-Maserrat, Discrete Applied Mathematics, 2005]. In this paper, we investigate the complexity of finding minimum forcing and anti-forcing sets for the shortest $s$-$t$ path problem and the minimum-weight spanning tree problem. We show that, unlike the aforementioned results, these problems are tractable, with the exception of the decision version of finding a minimum anti-forcing set for shortest $s$-$t$ paths, which is NP-complete. To complement this intractability result, we design fixed-parameter tractable algorithms for finding a minimum anti-forcing set for shortest $s$-$t$ paths.

cs.DS↗

arXiv2025

Hitting Geodesic Intervals in Structurally Restricted Graphs

Given a graph $G = (V,E)$, a set $T$ of vertex pairs, and an integer $k$, Hitting Geodesic Intervals asks whether there is a set $S \subseteq V$ of size at most $k$ such that for each terminal pair $\{u,v\} \in T$, the set $S$ intersects at least one shortest $u$-$v$ path. Aravind and Saxena [WALCOM 2024] introduced this problem and showed several parameterized complexity results. In this paper, we extend the known results in both negative and positive directions and present sharp complexity contrasts with respect to structural graph parameters. We first show that the problem is NP-complete even on graphs obtained by adding a single vertex to a disjoint union of 5-vertex paths. By modifying the proof of this result, we also show the NP-completeness on graphs obtained from a path by adding one vertex and on graphs obtained from a disjoint union of triangles by adding one universal vertex. Furthermore, we show the NP-completeness on graphs of bandwidth 4 and maximum degree 5 by replacing the universal vertex in the last case with a long path. Under standard complexity assumptions, these negative results rule out fixed-parameter algorithms for most of the structural parameters studied in the literature (if the solution size $k$ is not part of the parameter). We next present fixed-parameter algorithms parameterized by $k$ plus modular-width and by $k$ plus vertex integrity. The algorithm for the latter case does indeed solve a more general setting that includes the parameterization by the minimum vertex multiway-cut size of the terminal vertices. We show that this is tight in the sense that the problem parameterized by the minimum vertex multicut size of the terminal pairs is W[2]-complete. We then modify the proof of this intractability result and show that the problem is W[2]-complete parameterized by $k$ even in the setting where $T = \binom{Q}{2}$ for some $Q \subseteq V$.

cs.DS↗

arXiv2025

2-Layer Fan-Planarity in Polynomial Time

In this paper, we give a polynomial-time algorithm for deciding whether an input bipartite graph admits a 2-layer fan-planar drawing, resolving an open problem posed in several papers since 2015.

cs.CG↗

arXiv2025

Pathwidth of 2-Layer $k$-Planar Graphs

A bipartite graph $G = (X \cup Y, E)$ is a 2-layer $k$-planar graph if it admits a drawing on the plane such that the vertices in $X$ and $Y$ are placed on two parallel lines respectively, edges are drawn as straight-line segments, and every edge involves at most $k$ crossings. Angelini, Da Lozzo, F\"orster, and Schneck [GD 2020; Comput. J., 2024] showed that every 2-layer $k$-planar graph has pathwidth at most $k + 1$. In this paper, we show that this bound is sharp by giving a 2-layer $k$-planar graph with pathwidth $k + 1$ for every $k \geq 0$. This improves their lower bound of $(k+3)/2$.

cs.DM↗

arXiv2025

Structural Parameterizations of $k$-Planarity

The concept of $k$-planarity is extensively studied in the context of Beyond Planarity. A graph is $k$-planar if it admits a drawing in the plane in which each edge is crossed at most $k$ times. The local crossing number of a graph is the minimum integer $k$ such that it is $k$-planar. The problem of determining whether an input graph is $1$-planar is known to be NP-complete even for near-planar graphs [Cabello and Mohar, SIAM J. Comput. 2013], that is, the graphs obtained from planar graphs by adding a single edge. Moreover, the local crossing number is hard to approximate within a factor $2 - \varepsilon$ for any $\varepsilon > 0$ [Urschel and Wellens, IPL 2021]. To address this computational intractability, Bannister, Cabello, and Eppstein [JGAA 2018] investigated the parameterized complexity of the case of $k = 1$, particularly focusing on structural parameterizations on input graphs, such as treedepth, vertex cover number, and feedback edge number. In this paper, we extend their approach by considering the general case $k \ge 1$ and give (tight) parameterized upper and lower bound results. In particular, we strengthen the aforementioned lower bound results to subclasses of constant-treewidth graphs: we show that testing $1$-planarity is NP-complete even for near-planar graphs with feedback vertex set number at most $3$ and pathwidth at most $4$, and the local crossing number is hard to approximate within any constant factor for graphs with feedback vertex set number at most $2$.

cs.DS↗

arXiv2024

Recognizing 2-Layer and Outer $k$-Planar Graphs

The crossing number of a graph is the least number of crossings over all drawings of the graph in the plane. Computing the crossing number of a given graph is NP-hard, but fixed-parameter tractable (FPT) with respect to the natural parameter. Two well-known variants of the problem are 2-layer crossing minimization and circular crossing minimization, where every vertex must lie on one of two layers, namely two parallel lines, or a circle, respectively. Both variants are NP-hard, but FPT with respect to the natural parameter. Recently, a local version of the crossing number has also received considerable attention. A graph is $k$-planar if it admits a drawing with at most $k$ crossings per edge. In contrast to the crossing number, recognizing $k$-planar graphs is NP-hard even if $k=1$. In this paper, we consider the two above variants in the local setting. The $k$-planar graphs that admit a straight-line drawing with vertices on two layers or on a circle are called 2-layer $k$-planar and outer $k$-planar graphs, respectively. We study the parameterized complexity of the two recognition problems with respect to $k$. For $k=0$, both problems can easily be solved in linear time. Two groups independently showed that outer 1-planar graphs can also be recognized in linear time [Hong et al., Algorithmica 2015; Auer et al., Algorithmica 2016]. One group asked whether outer 2-planar graphs can be recognized in polynomial time. Our main contribution consists of XP-algorithms for recognizing 2-layer $k$-planar graphs and outer $k$-planar graphs. We complement these results by showing that both recognition problems are XNLP-hard. This implies that both problems are W$[t]$-hard for every $t$ and that it is unlikely that they admit FPT-algorithms. On the other hand, we present an FPT-algorithm for recognizing 2-layer $k$-planar graphs where the order of the vertices on one layer is specified.

cs.DS↗

arXiv2024

Basis sequence reconfiguration in the union of matroids

Given a graph $G$ and two spanning trees $T$ and $T'$ in $G$, Spanning Tree Reconfiguration asks whether there is a step-by-step transformation from $T$ to $T'$ such that all intermediates are also spanning trees of $G$, by exchanging an edge in $T$ with an edge outside $T$ at a single step. This problem is naturally related to matroid theory, which shows that there always exists such a transformation for any pair of $T$ and $T'$. Motivated by this example, we study the problem of transforming a sequence of spanning trees into another sequence of spanning trees. We formulate this problem in the language of matroid theory: Given two sequences of bases of matroids, the goal is to decide whether there is a transformation between these sequences. We design a polynomial-time algorithm for this problem, even if the matroids are given as basis oracles. To complement this algorithmic result, we show that the problem of finding a shortest transformation is NP-hard to approximate within a factor of $c \log n$ for some constant $c > 0$, where $n$ is the total size of the ground sets of the input matroids.

math.CO↗

arXiv2024

Bounding the Treewidth of Outer $k$-Planar Graphs via Triangulations

The treewidth is a structural parameter that measures the tree-likeness of a graph. Many algorithmic and combinatorial results are expressed in terms of the treewidth. In this paper, we study the treewidth of outer $k$-planar graphs, that is, graphs that admit a straight-line drawing where all the vertices lie on a circle, and every edge is crossed by at most $k$ other edges. Wood and Telle [New York J. Math., 2007] showed that every outer $k$-planar graph has treewidth at most $3k + 11$ using so-called planar decompositions, and later, Auer et al. [Algorithmica, 2016] proved that the treewidth of outer $1$-planar graphs is at most $3$, which is tight. In this paper, we improve the general upper bound to $1.5k + 2$ and give a tight bound of $4$ for $k = 2$. We also establish a lower bound: we show that, for every even $k$, there is an outer $k$-planar graph with treewidth $k+2$. Our new bound immediately implies a better bound on the cop number, which answers an open question of Durocher et al. [GD 2023] in the affirmative. Our treewidth bound relies on a new and simple triangulation method for outer $k$-planar graphs that yields few crossings with graph edges per edge of the triangulation. Our method also enables us to obtain a tight upper bound of $k + 2$ for the separation number of outer $k$-planar graphs, improving an upper bound of $2k + 3$ by Chaplick et al. [GD 2017]. We also consider outer min-$k$-planar graphs, a generalization of outer $k$-planar graphs, where we achieve smaller improvements.

cs.DM↗

arXiv2023

Dichotomies for Tree Minor Containment with Structural Parameters

The problem of determining whether a graph $G$ contains another graph $H$ as a minor, referred to as the minor containment problem, is a fundamental problem in the field of graph algorithms. While it is NP-complete when $G$ and $H$ are general graphs, it is sometimes tractable on more restricted graph classes. This study focuses on the case where both $G$ and $H$ are trees, known as the tree minor containment problem. Even in this case, the problem is known to be NP-complete. In contrast, polynomial-time algorithms are known for the case when both trees are caterpillars or when the maximum degree of $H$ is a constant. Our research aims to clarify the boundary of tractability and intractability for the tree minor containment problem. Specifically, we provide dichotomies for the computational complexities of the problem based on three structural parameters: the diameter, pathwidth, and path eccentricity.

cs.DS↗

arXiv2022

Sequentially Swapping Tokens: Further on Graph Classes

We study the following variant of the 15 puzzle. Given a graph and two token placements on the vertices, we want to find a walk of the minimum length (if any exists) such that the sequence of token swappings along the walk obtains one of the given token placements from the other one. This problem was introduced as Sequential Token Swapping by Yamanaka et al. [JGAA 2019], who showed that the problem is intractable in general but polynomial-time solvable for trees, complete graphs, and cycles. In this paper, we present a polynomial-time algorithm for block-cactus graphs, which include all previously known cases. We also present general tools for showing the hardness of the problem on restricted graph classes such as chordal graphs and chordal bipartite graphs. We also show that the problem is hard on grids and king's graphs, which are the graphs corresponding to the 15 puzzle and its variant with relaxed moves.

cs.DS↗