Contact Graphs of Circular Arcs

Abstract

We study representations of graphs by contacts of circular arcs, CCA-representations for short, where the vertices are interiordisjoint circular arcs in the plane and each edge is realized by an endpoint of one arc touching the interior of another. A graph is (2, k)-sparse if every s-vertex subgraph has at most 2s − k edges, and (2, k)-tight if in addition it has exactly 2n − k edges, where n is the number of vertices. Every graph with a CCA-representation is planar and (2, 0)-sparse, and it follows from known results that for k ≥ 3 every (2, k)-sparse graph has a CCA-representation. Hence the question of CCA-representability is open for (2, k)-sparse graphs with 0 ≤ k ≤ 2. We partially answer this question by computing CCA-representations for several subclasses of planar (2, 0)-sparse graphs. Next, we study CCA-representations in which each arc has an empty convex hull. We show that every plane graph of maximum degree 4 has such a representation, but that finding such a representation for a plane (2, 0)-tight graph with maximum degree 5 is NP-complete. Finally, we describe a simple algorithm for representing plane (2, 0)-sparse graphs with wedges, where each vertex is represented with a sequence of two circular arcs (straight-line segments).

Triangle-Free Penny Graphs: Degeneracy, Choosability, and Edge Count

“…Penny graphs are the contact graphs of unit circles [1,2] -they are formed from non-overlapping sets of unit circles by creating a vertex for each circle and an edge for each tangency between two circles -and as such, fit into a long line of graph drawing research on contact graphs of geometric objects [3][4][5][6][7]. The same graphs (except the graph with no edges) are also proximity graphs, the graphs determined from a finite set of points in the plane by adding edges between all closest pairs of points, and for this reason they are also called minimum-distance graphs [8,9].…”