Optimal geometric representations of graphs and triangles

Abstract

We investigate three problems: obstacle representation of graphs, computing the bundled crossing number of a graph, and universal covers for families of triangles. First, we study single-obstacle graphs: visibility graphs that can be represented by placing vertices as points in the plane and a simple polygonal obstacle (vertices are adjacent if and only if they can see each other). An obstacle is called an outside-obstacle if it lies in the unbounded component of the visibility drawing and an inside-obstacle otherwise. We establish several subclasses of single-obstacle graphs and find a smallest graph that is not a single-obstacle graph. We show that inside-obstacle graphs and outside-obstacle graphs are incomparable. We prove that the single-obstacle graph sandwich recognition problem and the simple-polygon visibility graph sandwich recognition problem are NP-hard. Second, we study a technique to reduce the visual complexity of a graph drawing by bundling locally parallel edges, where we count the number of bundled crossings and call its minimum over all simple drawings the bundled crossing number. We show that computing the bundled crossing number is NP-complete. We also show that computing several variants of the circular bundled crossing number where vertices are placed on a circle and edges are drawn inside the circle are fixed-parameter tractable. Third, given a family of triangles, we want to find a convex cover of the smallest area that can accommodate a congruent copy of every triangle. We conjecture that the answer is a triangle for any set of triangles, and show that this conjecture is equivalent to the conjecture that the answer is a triangle for the family of triangles contained in any given convex set. We demonstrate that the answer is indeed a triangle for two cases, namely (i) triangles of unit circumradius and (ii) any two triangles.