On Representation of Planar Graphs by Segments

Abstract

In this paper, we introduce Vertex-face contact representation (VFCR for short) for 2-connected plane multigraphs. We present a simple linear time algorithm for constructing a VFCR for 2-connected plane graphs. Our algorithm only uses an st-orientation for G and its corresponding st-orientation for the dual graph of G. We also show that one kind of vertex-vertex contact representation (VVCR) for 2-connected bipartite planar graphs introduced by Fraysseix et al. [2,3] can be easily obtained by applying our algorithm. In general, our algorithm produces a more compact representation than their algorithm.Then we investigate st-orientations for 3-connected planar graphs. We prove that a 3-connected planar graph G with n vertices and f faces, has an st-orientation with the length of its longest directed path \(\leq \frac{2n}{3}+2\lceil\sqrt{n/3}\rceil+5\). This implies that such a graph G admits a VFCR in a grid with non-trivial size bound. This non-trivial size bound also applies to the vertex-vertex contact representation [2,3] for a large class of 2-connected bipartite planar graphs.