Finding Hamiltonian cycles in Delaunay triangulations is NP-complete

AbstractIt is shown that it is an NP-complete problem to determine whether a Delaunay triangulation or an inscribable polyhedron has a Hamiltonian cycle. It is also shown that there exist nondegenerate Delaunay triangulations and simplicial, inscribable polyhedra without 2-factors

Square-Contact Representations of Partial 2-Trees and Triconnected Simply-Nested Graphs

A square-contact representation of a planar graph G = (V,E) maps vertices in V to interior-disjoint axis-aligned squares in the plane and edges in E to adjacencies between the sides of the corresponding squares. In this paper, we study proper square-contact representations of planar graphs, in which any two squares are either disjoint or share infinitely many points. We characterize the partial 2-trees and the triconnected cycle-trees allowing for such representations. For partial 2-trees our characterization uses a simple forbidden subgraph whose structure forces a separating triangle in any embedding. For the triconnected cycle-trees, a subclass of the triconnected simply-nested graphs, we use a new structural decomposition for the graphs in this family, which may be of independent interest. Finally, we study square-contact representations of general triconnected simply-nested graphs with respect to their outerplanarity index

Subexponential-Time and FPT Algorithms for Embedded Flat Clustered Planarity

The C-Planarity problem asks for a drawing of a $\textit{clustered graph}$, i.e., a graph whose vertices belong to properly nested clusters, in which each cluster is represented by a simple closed region with no edge-edge crossings, no region-region crossings, and no unnecessary edge-region crossings. We study C-Planarity for $\textit{embedded flat clustered graphs}$, graphs with a fixed combinatorial embedding whose clusters partition the vertex set. Our main result is a subexponential-time algorithm to test C-Planarity for these graphs when their face size is bounded. Furthermore, we consider a variation of the notion of $\textit{embedded tree decomposition}$ in which, for each face, including the outer face, there is a bag that contains every vertex of the face. We show that C-Planarity is fixed-parameter tractable with the embedded-width of the underlying graph and the number of disconnected clusters as parameters.Comment: 14 pages, 6 figure

Planar Drawings with Few Slopes of Halin Graphs and Nested Pseudotrees

The $\textit{planar slope number}$ $psn(G)$ of a planar graph $G$ is the minimum number of edge slopes in a planar straight-line drawing of $G$. It is known that $psn(G) \in O(c^\Delta)$ for every planar graph $G$ of maximum degree $\Delta$. This upper bound has been improved to $O(\Delta^5)$ if $G$ has treewidth three, and to $O(\Delta)$ if $G$ has treewidth two. In this paper we prove $psn(G) \leq \max\{4,\Delta\}$ when $G$ is a Halin graph, and thus has treewidth three. Furthermore, we present the first polynomial upper bound on the planar slope number for a family of graphs having treewidth four. Namely we show that $O(\Delta^2)$ slopes suffice for nested pseudotrees.Comment: Extended version of "Planar Drawings with Few Slopes of Halin Graphs and Nested Pseudotrees" appeared in the Proceedings of the 17th Algorithms and Data Structures Symposium (WADS 2021