arXiv subjects

Thomas Depian

Publications and source records attributed to Thomas Depian.

15 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

Point Set Embeddability with List Constraints

Deciding whether a given graph admits a planar straight-line drawing where each vertex is placed on some point from a given finite point set is known as Point Set Embeddability and is a classical problem in graph drawing. In this paper, we study the more general embeddability question where the placement of each vertex $v$ is restricted to a list $L(v)$ of admissible points. We first study the case where the given point set is in convex position. We show that this case is NP-hard even if the given graph is a matching and bi-labeled, i.e., each vertex has at most 2 admissible points. On the positive side, we present two efficient algorithms for the case where the given graph $G$ is connected (and not necessarily bi-labeled): if $G$ is equipped with a combinatorial embedding that needs to be respected, we can solve the problem in polynomial time; otherwise we can solve it in FPT-time with regard to the maximum vertex degree. In particular, this answers an open question by Frati, Glisse, Lenhart, Liotta, Mchedlidze, and Nishat [GD'13]. We then turn our attention to the more general case where the given point set is not necessarily in convex position. Here, we show NP-hardness for bi-labeled paths; notably these graphs have a unique combinatorial embedding and maximum degree two. We also present an FPT-algorithm with respect to the vertex cover number for the special case of bi-labeled graphs. We complement this latter result by establishing paraNP-hardness in the tri-labeled setting for vertex cover number 2 and polynomial-time solvability for vertex cover number 1 and arbitrary $L$. Finally, we study optimization and extension variants, where we want to maximize the number of edges or extend a partial drawing, respectively. For the former, we show APX-hardness and for the latter, we provide a parameterized complexity dichotomy under natural extension parameters.

cs.CG↗

arXiv2026

On the Recognition of Outerplanar Graphs with Queue Number 1

A linear layout of a graph is defined as a total order of the vertices and a partition of the edges to pages. In a stack (queue) layout, no two edges on the same page may cross (nest). The stack (queue) number of a graph is the minimum number of pages required in a stack (queue) layout. This paper focuses on characterizing and recognizing graphs that have both stack number 1 and queue number 1. It is known that the graphs with stack number 1 are exactly the outerplanar graphs. We show that (i) deciding whether a given outerplanar graph has queue number 1 is NP-hard; (ii) deciding whether a given maximal outerplanar graph has queue number 1 can be done in linear time. Moreover, we investigate the interplay between outerpaths with queue number 1 and their maximum vertex degree.

cs.CG↗

arXiv2026

Paged Geophylogenies: A Coloring Approach to External Labeling with Tree Constraints

Geophylogenies are a common type of diagram for visualizing the evolutionary history of species in a geographic context. As a drawing problem, these diagrams are commonly modeled as a rooted binary ''phylogenetic'' tree $T$ where every leaf is associated with a point feature (''site'') in a rectangular map range. The tree is drawn downward planar, its leaves are placed at equidistant positions on the upper boundary of the map, and each leaf is connected to its corresponding site by a straight-line leader. Prior work focuses on minimizing the number of leader crossings, or avoiding leaders altogether. In this paper, we explore paged geophylogenies, where the leaders can be partitioned into multiple pages where only crossings within a page are counted. For the general case, where each page can contain an arbitrary subset of the leaders, we provide an integer linear programming (ILP) formulation for minimizing the number of pages, and a polynomial-time algorithm for a special case that is equivalent to a one-sided tanglegram. We argue that, from a visualization perspective, instead each page must contain only leaders for a single subtree, and provide an $\mathcal{O}(n^7)$ time algorithm for minimizing pages in this setting - which also involves improving the fastest known algorithm for testing the existence of a crossing-free labeling. Counter to what our worst case bound suggests, our implementation can handle instances with hundreds of sites within a second, as we show in our experimental evaluation, which further investigates the trade-offs between leader crossings and the number of pages.

cs.CG↗

arXiv2026

Realizing Planar Linkages in Polygonal Domains

A linkage $\mathcal{L}$ consists of a graph $G=(V,E)$ and an edge-length function $\ell$. Deciding whether $\mathcal{L}$ can be realized as a planar straight-line embedding in $\mathbb{R}^2$ with edge length $\ell(e)$ for all $e \in E$ is $\exists\mathbb{R}$-complete [Abel et al., JoCG'25], even if $\ell \equiv 1$, but a considerable part of $\mathcal{L}$ is rigid. In this paper, we study the computational complexity of the realization question for structurally simpler, less rigid linkages inside an open polygonal domain $P$, where the placement of some vertices may be specified in the input. We show XP-membership and W[1]-hardness with respect to the size of $G$, even if $\ell \equiv 1$ and no vertex positions are prescribed. Furthermore, we consider the case where $G$ is a path with prescribed start and end position and $\ell \equiv 1$. Despite the absence of any rigid components, we obtain NP-hardness in general, and provide a linear-time algorithm for arbitrary $\ell$ if $G$ has only three edges and $P$ is convex.

cs.CG↗

arXiv2026

Revisiting Graph Modification via Disk Scaling: From One Radius to Interval-Based Radii

For a fixed graph class $\Pi$, the goal of $\Pi$-Modification is to transform an input graph $G$ into a graph $H\in\Pi$ using at most $k$ modifications. Vertex and edge deletions are common operations, and their (parameterized) complexity for various $\Pi$ is well-studied. Classic graph modification operations such as edge deletion do not consider the geometric nature of intersection graphs such as (unit) disk graphs. This led Fomin et al. [ITCS' 25] to introduce scaling as a geometric graph modification operation for unit disk graphs: For a given radius $r$, each modified disk will be rescaled to radius $r$. In this paper, we generalize their model by allowing rescaled disks to choose a radius within a given interval $[r_{\min}, r_{\max}]$ and study the (parameterized) complexity (with respect to $k$) of the corresponding problem $\Pi$-Scaling. We show that $\Pi$-Scaling is in XP for every graph class $\Pi$ that can be recognized in polynomial time. Furthermore, we show that $\Pi$-Scaling: (1) is NP-hard and FPT for cluster graphs, (2) can be solved in polynomial time for complete graphs, and (3) is W[1]-hard for connected graphs. In particular, (1) and (2) answer open questions of Fomin et al. and (3) generalizes the hardness result for their variant where the set of scalable disks is restricted.

cs.CG↗

arXiv2025

Visualizing Treewidth

A witness drawing of a graph is a visualization that clearly shows a given property of a graph. We study and implement various drawing paradigms for witness drawings to clearly show that graphs have bounded pathwidth or treewidth. Our approach draws the tree decomposition or path decomposition as a tree of bags, with induced subgraphs shown in each bag, and with ''tracks'' for each vertex of the graph connecting its copies in multiple bags. Within bags, we optimize the vertex layout to avoid crossings of edges and tracks. We implement a visualization prototype for crossing minimization using dynamic programming for graphs of small width and heuristic approaches for graphs of larger width. We explore the design space for width-witness drawings and investigate drawing styles that render the subgraph for each bag as an arc diagram with one or two pages or as a circular layout with straight-line edges, and we render tracks either with straight lines or with orbital-radial paths. Finally, we report results from an expert evaluation assessing different witness drawing styles.

cs.CG↗

arXiv2025

Linear Layouts Revisited: Stacks, Queues, and Exact Algorithms

In spite of the extensive study of stack and queue layouts, many fundamental questions remain open concerning the complexity-theoretic frontiers for computing stack and queue layouts. A stack (resp. queue) layout places vertices along a line and assigns edges to pages so that no two edges on the same page are crossing (resp. nested). We provide three new algorithms which together substantially expand our understanding of these problems: (1) A fixed-parameter algorithm for computing minimum-page stack and queue layouts w.r.t. the vertex integrity of an n-vertex graph G. This result is motivated by an open question in the literature and generalizes the previous algorithms parameterizing by the vertex cover number of G. The proof relies on a newly developed Ramsey pruning technique. Vertex integrity intuitively measures the vertex deletion distance to a subgraph with only small connected components. (2) An n^(O(q * l)) algorithm for computing l-page stack and queue layouts of page width at most q. This is the first algorithm avoiding a double-exponential dependency on the parameters. The page width of a layout measures the maximum number of edges one needs to cross on any page to reach the outer face. (3) A 2^(O(n)) algorithm for computing 1-page queue layouts. This improves upon the previously fastest n^(O(n)) algorithm and can be seen as a counterpart to the recent subexponential algorithm for computing 2-page stack layouts [ICALP'24], but relies on an entirely different technique.

cs.DS↗

arXiv2025

The Peculiarities of Extending Queue Layouts

We consider the problem of computing $\ell$-page queue layouts, which are linear arrangements of vertices accompanied with an assignment of the edges to pages from one to $\ell$ that avoid the nesting of edges on any of the pages. Inspired by previous work in the extension of stack layouts, here we consider the setting of extending a partial $\ell$-page queue layout into a complete one and primarily analyze the problem through the refined lens of parameterized complexity. We obtain novel algorithms and lower bounds which provide a detailed picture of the problem's complexity under various measures of incompleteness, and identify surprising distinctions between queue and stack layouts in the extension setting.

cs.CG↗

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

The Parameterized Complexity of Extending Stack Layouts

An $\ell$-page stack layout (also known as an $\ell$-page book embedding) of a graph is a linear order of the vertex set together with a partition of the edge set into $\ell$ stacks (or pages), such that the endpoints of no two edges on the same stack alternate. We study the problem of extending a given partial $\ell$-page stack layout into a complete one, which is a natural generalization of the classical NP-hard problem of computing a stack layout of an input graph from scratch. Given the inherent intractability of the problem, we focus on identifying tractable fragments through the refined lens of parameterized-complexity analysis. Our results paint a detailed and surprisingly rich complexity-theoretic landscape of the problem which includes the identification of paraNP-hard, W[1]-hard and XP-tractable, as well as fixed-parameter tractable fragments of stack layout extension via a natural sequence of parameterizations.

cs.CG↗

arXiv2024

Constrained Boundary Labeling

Boundary labeling is a technique in computational geometry used to label sets of features in an illustration. It involves placing labels along an axis-parallel bounding box and connecting each label with its corresponding feature using non-crossing leader lines. Although boundary labeling is well-studied, semantic constraints on the labels have not been investigated thoroughly. In this paper, we introduce grouping and ordering constraints in boundary labeling: Grouping constraints enforce that all labels in a group are placed consecutively on the boundary, and ordering constraints enforce a partial order over the labels. We show that it is NP-hard to find a labeling for arbitrarily sized labels with unrestricted positions along one side of the boundary. However, we obtain polynomial-time algorithms if we restrict this problem either to uniform-height labels or to a finite set of candidate positions. Furthermore, we show that finding a labeling on two opposite sides of the boundary is NP-complete, even for uniform-height labels and finite label positions. Finally, we experimentally confirm that our approach has also practical relevance.

cs.CG↗

arXiv2023

Minimizing Corners in Colored Rectilinear Grids

Given a rectilinear grid $G$, in which cells are either assigned a single color, out of $k$ possible colors, or remain white, can we color white grid cells of $G$ to minimize the total number of corners of the resulting colored rectilinear polygons in $G$? We show how this problem relates to hypergraph visualization, prove that it is NP-hard even for $k=2$, and present an exact dynamic programming algorithm. Together with a set of simple kernelization rules, this leads to an FPT-algorithm in the number of colored cells of the input. We additionally provide an XP-algorithm in the solution size, and a polynomial $\mathcal{O}(OPT)$-approximation algorithm.

cs.CG↗

arXiv2023

Network Navigation with Online Delays is PSPACE-complete

In public transport networks disruptions may occur and lead to travel delays. It is thus interesting to determine whether a traveler can be resilient to delays that occur unexpectedly, ensuring that they can reach their destination in time regardless. We model this as a game between the traveler and a delay-introducing adversary. We study the computational complexity of the problem of deciding whether the traveler has a winning strategy in this game. Our main result is that this problem is PSPACE-complete.

cs.CC↗

arXiv2022

Transitions in Dynamic Point Labeling

The labeling of point features on a map is a well-studied topic. In a static setting, the goal is to find a non-overlapping label placement for (a subset of) point features. In a dynamic setting, the set of point features and their corresponding labels change, and the labeling has to adapt to such changes. To aid the user in tracking these changes, we can use morphs, here called transitions, to indicate how a labeling changes. Such transitions have not gained much attention yet, and we investigate different types of transitions for labelings of points, most notably consecutive transitions and simultaneous transitions. We give (tight) upper bounds on the number of overlaps that can occur during these transitions. When each label has a non-negative weight associated to it, and each overlap imposes a penalty proportional to the weight of the overlapping labels, we show that it is NP-complete to decide whether the penalty during a simultaneous transition has weight at most $k$. Finally, we consider geotagged data on a map, by labeling points with rectangular or square labels. We developed a prototype implementation to evaluate different transition styles in practice, measuring both number of overlaps and transition duration.

cs.CG↗