We provide a new linear-time algorithm for determining the vertex-connectivity of graphs with bounded genus. This generalizes and streamlines a linear-time algorithm for graphs with bounded crossing number which was recently obtained by Biedl, Bose and Murali [ESA 2024]. Compared to applying the even more recent fixed parameter linear-time algorithm for deciding bounded vertex-connectivity announced by Korhonen [STOC 2025] to graphs of bounded genus,our algorithm is far simpler, its correctness easier to establish, and it makes use of geometric ideas, as is natural for surface-embedded graphs.

Determining the crossing numbers of Cartesian products of small graphs with arbitrarily large paths has been an ongoing topic of research since the 1970s. Doing so requires the establishment of coincident upper and lower bounds; the former is usually demonstrated by providing a suitable drawing procedure, while the latter often requires substantial theoretical arguments. Many such papers have been published, which typically focus on just one or two small graphs at a time, and use ad hoc arguments specific to those graphs. We propose a general approach which, when successful, establishes the required lower bound. This approach can be applied to the Cartesian product of any graph with arbitrarily large paths, and in each case involves solving a modified version of the crossing number problem on a finite number (typically only two or three) of small graphs. We demonstrate the potency of this approach by applying it to Cartesian products involving all 133 graphs of orders five or six, and show that it is successful in 128 cases. This includes 60 cases which a recent survey listed as either undetermined, or determined only in journals without adequate peer review.

A drawing of a graph is 1-planar if each edge participates in at most one crossing and adjacent edges do not cross. Up to symmetry, each crossing in a 1-planar drawing belongs to one out of six possible crossing types, where a type characterizes the subgraph induced by the four vertices of the crossing edges. Each of the 63 possible nonempty subsets S of crossing types gives a recognition problem: does a given graph admit an S-restricted drawing, that is, a 1-planar drawing where the crossing type of each crossing is in S? We show that there is a set Sbad with three crossing types and the following properties: If S contains no crossing type from Sbad, then the recognition of graphs that admit an S-restricted drawing is fixed-parameter tractable with respect to the treewidth of the input graph. If S contains any crossing type from Sbad, then it is NP-hard to decide whether a graph has an S-restricted drawing, even when considering graphs of constant pathwidth. We also extend this characterization of crossing types to 1-planar straight-line drawings and show the same complexity behaviour parameterized by treewidth.

Collaboration among drone swarms in various applications often requires dynamic data exchange to optimize their missions. When employing device-to-device (D2D) communications, drones must form connected topologies enabling multi-hop communication. Maintaining a connected topology during their movement to new locations is a major challenge that has not been tackled in the literature. In this paper, we first formalize this problem, proving the decision version is NP-Hard, and break it into sub-problems, detailing their complexity under certain constraints. We then propose polynomial time solutions to these subproblems. Our proposed approach first makes a location assignment for each drone to determine where to move as their next locations. Next, our approach relocates each drone to their designated target location without losing in-network connectivity of the drone topology. We achieve this by leveraging a two-step approach: contraction and re-expansion. In the contraction phase, drones converge towards a specific location, reducing network diameter and increasing connectivity. In the expansion phase, drones disperse to their new target positions. Mathematical proofs and extensive simulations validate our method’s effectiveness in maintaining connectivity during drone relocations.

Given a weighted graph $G$, a minimum weight $α$-spanner is a least-weight subgraph $H\subseteq G$ that preserves minimum distances between all node pairs up to a factor of $α$. There are many results on heuristics and approximation algorithms, including a recent investigation of their practical performance [20]. Exact approaches, in contrast, have long been denounced as impractical: The first exact ILP (integer linear program) method [48] from 2004 is based on a model with exponentially many path variables, solved via column generation. A second approach [2], modeling via arc-based multicommodity flow, was presented in 2019. In both cases, only graphs with 40-100 nodes were reported to be solvable. In this paper, we briefly report on a theoretical comparison between these two models from a polyhedral point of view, and then concentrate on improvements and engineering aspects. We evaluate their performance in a large-scale empirical study. We report that our tuned column generation approach, based on multicriteria shortest path computations, is able to solve instances with over 16000 nodes within 13 minutes. Furthermore, now knowing optimal solutions for larger graphs, we are able to investigate the quality of the strongest known heuristic on reasonably sized instances for the first time.

Visualizing a graph $G$ in the plane nicely, for example, without crossings, is unfortunately not always possible. To address this problem, Masařík and Hliněný [GD 2023] recently asked for each edge of $G$ to be drawn without crossings while allowing multiple different drawings of $G$. More formally, a collection $\mathcal{D}$ of drawings of $G$ is uncrossed if, for each edge $e$ of $G$, there is a drawing in $\mathcal{D}$ such that $e$ is uncrossed. The uncrossed number $\mathrm{unc}(G)$ of $G$ is then the minimum number of drawings in some uncrossed collection of $G$. No exact values of the uncrossed numbers have been determined yet, not even for simple graph classes. In this paper, we provide the exact values for uncrossed numbers of complete and complete bipartite graphs, partly confirming and partly refuting a conjecture posed by Hliněný and Masařík. We also present a strong general lower bound on $\mathrm{unc}(G)$ in terms of the number of vertices and edges of $G$. Moreover, we prove NP-hardness of the related problem of determining the edge crossing number of a graph $G$, which is the smallest number of edges of $G$ taken over all drawings of $G$ that participate in a crossing. This problem was posed as open by Schaefer in his book [Crossing Numbers of Graphs 2018].

Beyond planarity concepts (prominent examples include k-planarity or fan-planarity) apply certain restrictions on the allowed patterns of crossings in drawings. It is natural to ask, how much the number of crossings may increase over the traditional (unrestricted) crossing number. Previous approaches to bound such ratios, e.g. [arXiv:1908.03153, arXiv:2105.12452], require very specialized constructions and arguments for each considered beyond planarity concept, and mostly only yield asymptotically non-tight bounds. We propose a very general proof framework that allows us to obtain asymptotically tight bounds, and where the concept-specific parts of the proof typically boil down to a couple of lines. We show the strength of our approach by giving improved or first bounds for several beyond planarity concepts.