Alexander Wolff

Steven Chaplick, Myroslav Kryven, Giuseppe Liotta et al.

We study straight-line drawings of graphs where the vertices are placed in convex position in the plane, i.e., \emph{convex drawings}. We consider two families of graph classes with convex drawings: \emph{outer $k$-planar} graphs, where each edge is crossed by at most $k$ other edges; and \emph{outer $k$-quasi-planar} graphs, where no $k$ edges can mutually cross. We show that the outer $k$-planar graphs are $\lfloor3.5\sqrt{k}\rfloor$-degenerate, and consequently that every outer $k$-planar graph can be colored with $\lfloor3.5\sqrt{k}\rfloor + 1$ colors. We further show that every outer $k$-planar graph has a balanced vertex separator of size at most $2k+3$. For each fixed $k$, these small balanced separators allow us to test outer $k$-planarity in quasi-polynomial time, e.g., this implies that none of these recognition problems is NP-hard unless the Exponential Time Hypothesis fails. We also show that the class of outer 3-quasi-planar graphs and the class of planar graphs are incomparable. Finally, we restrict outer $k$-planar and outer $k$-quasi-planar drawings to \emph{full} drawings (where no crossing appears on the boundary of the outer face) and to \emph{closed} drawings (where the vertex sequence on the boundary of the outer face is a Hamiltonian cycle in the graph). For each $k$, we express \emph{closed outer $k$-planarity} and \emph{closed outer $k$-quasi-planarity} in extended monadic second-order logic. Since every outer $k$-planar graph has treewidth $O(k)$, Courcelle's theorem implies that closed outer $k$-planarity is linear-time testable. We leverage this result to further show that full outer $k$-planarity can also be tested in linear time.

Nicolás Honorato-Droguett, Kazuhiro Kurita, Tesshu Hanaka et al.

Removing overlaps is a central task in domains such as scheduling, visibility, and map labelling. This can be modelled using graphs, where overlap removals correspond to enforcing a certain sparsity constraint on the graph structure. We continue the study of the problem Geometric Graph Edit Distance (GGED), where the aim is to minimise the total cost of editing a geometric intersection graph to obtain a graph contained in a specific graph class. For us, the edit operation is the movement of objects, and the cost is the movement distance. We present an algorithm for rendering the intersection graph of a set of unit circular arcs edgeless and $k$-clique-free in $O(n\log n)$ time, where $n$ is the number of arcs. The algorithm can be also used to solve an open case of the points-spreading problem on cyclic domains [Li \& Wang, CGT 2025]. We also show that GGED remains strongly NP-hard on unweighted interval graphs, solving an open problem of Honorato-Droguett et al. [WADS 2025]. We complement this result by showing that GGED is strongly NP-hard on sets of $d$-balls and $d$-cubes, for any $d\ge 2$. Finally, we present an XP algorithm (parameterised by the number of maximal cliques) that removes all edges from the intersection graph of a set of weighted unit intervals.

Antonio Lauerbach, Konstanty Junosza-Szaniawski, Marie Diana Sieper et al.

A mixed graph contains (undirected) edges as well as (directed) arcs, thus generalizing undirected and directed graphs. A proper coloring c of a mixed graph G assigns a positive integer to each vertex such that c(u)!=c(v) for every edge {u,v} and c(u)<c(v) for every arc (u,v) of G. As in classical coloring, the objective is to minimize the number of colors. Thus, mixed (graph) coloring generalizes classical coloring of undirected graphs and allows for more general applications, such as scheduling with precedence constraints, modeling metabolic pathways, and process management in operating systems; see a survey by Sotskov [Mathematics, 2020]. We initiate the systematic study of the parameterized complexity of mixed coloring. We focus on structural graph parameters that lie between cliquewidth and vertex cover, primarily with respect to the underlying undirected graph. Unlike classical coloring, which is fixed-parameter tractable (FPT) parameterized by treewidth or neighborhood diversity, we show that mixed coloring is W[1]-hard for treewidth and even paraNP-hard for neighborhood diversity. To utilize the directedness of arcs, we introduce and analyze natural generalizations of neighborhood diversity and cliquewidth to mixed graphs, and show that mixed coloring becomes FPT when parameterized by mixed neighborhood diversity. Further, we investigate how these parameters are affected if we add transitive arcs, which do not affect colorings. Finally, we provide tight bounds on the chromatic number of mixed graphs, generalizing known bounds on mixed interval graphs.

Grzegorz Gutowski, Konstanty Junosza-Szaniawski, Antonio Lauerbach et al.

Motivated by the controller placement problems in software-defined networks and the fair division principles of classical "cake cutting", we investigate the following two-player zero-sum game. In our model, a defender places a limited number of controllers on graph vertices, while an attacker deletes a limited number of vertices. The defender score is the total number of surviving vertices reachable from any remaining controller. We formalize the computational problems associated with various game dynamics (defender plays first; attacker plays first; players play simultaneously; pure or mixed strategies). We show that these natural problems are $\mathsf{NP}$-complete or $Σ^\mathsf{P}_2$-complete, depending on the specific variant. These hardness results provide limitations for optimal controller placement algorithms under different notions of quality of a solution. Finally, we present structural insights that yield efficient algorithms for restricted graph classes (namely interval graphs and graphs of bounded treewidth).

Panos Giannopoulos, Miriam Goetze, Grzegorz Gutowski et al.

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.

Todor Antić, Giuseppe Liotta, Tomáš Masařík et al.

Recently, there has been interest in representing single graphs by multiple drawings; for example, using graph stories, storyplans, or uncrossed collections. In this paper, we apply this idea to orthogonal graph drawing. Due to the orthogonal drawing style, we focus on 4-graphs, that is, graphs of maximum degree 4. We restrict ourselves to plane graphs, that is, planar graphs whose embedding is fixed. Our goal is to represent any plane 4-graph $G$ by an unbent collection, that is, a collection of orthogonal drawings of $G$ that adhere to the embedding of $G$ and ensure that each edge of $G$ is drawn without bends in at least one of the drawings. We investigate two objectives. First, we consider minimizing the number of drawings in an unbent collection. We prove that every plane 4-graph can be represented by a collection with at most three drawings, which is tight. We also give necessary and sufficient conditions for a graph to admit an unbent collection of size $2$. Second, we consider minimizing the total number of bends over all drawings in an unbent collection. We show that this problem is NP-hard and give a 3-approximation algorithm. For the special case of plane triconnected cubic graphs, we show how to compute minimum-bend collections in linear time.