On the 3-tree core of plane graphs

Abstract

A drawing of a graph is a geometric representation of its vertices and edges. Plane 3-trees have been well studied in graph drawing literature. For many graph drawing styles, the aesthetic qualities achieved for plane 3-trees are much better than the ones known for general plane graphs. This motivates us to investigate whether one can find a large plane 3-tree type structure in a general plane graph, and if so, whether it can be leveraged to obtain a better drawing for the graph. We thus introduce the concept of a 3-tree core H of a 3-connected plane graph G. Here, H is an edge-labeled plane 3-tree that represents G, and the distance d between H and G is the number of vertices of G that are missing in H. As an application of this concept, we consider the planar ortho-path visibility drawing, where each vertex is drawn as an orthogonal polygonal chain on an integer grid and each edge is drawn as an orthogonal line segment between the paths corresponding to its end vertices. We show that if H has a flat visibility drawing (i.e., each ortho-path is a horizontal line segment) with height k, then G has an ortho-path visibility drawing with height \(O(k2^d)\). In particular, if G is a planar triangulation and not too distant from a 3-tree core, i.e., \(d=O(1)\), then G can be drawn with height \(4n/9+O(1)\) by choosing an appropriate planar embedding. This bound is interesting as it is significantly smaller than the lower bound of \(2n/3+O(1)\) when the ortho-path visibility drawing must respect the input embedding.