Proximity drawings of outerplanar graphs

Abstract

A proximity drawing of a graph is one in which pairs of adjacent vertices axe drawn relatively close together according to some proximity measure while pairs of non-adjacent vertices are drawn relatively far apart. The fundamental question concerning proximity drawability is: Given a graph G and a definition of proximity, is it possible to construct a proximity drawing of G? We consider this question for outerplanar graphs with respect to an infinite family of proximity drawings called β-drawings. These drawings include as special cases the well-known Gabriel drawings (when β = 1), and relative neighborhood drawings (when β = 2). We first show that all biconnected outerplanar graphs are β-drawable for all valnes of β such that 1 ≤ β ≤ 2. As a side effect, this result settles in the affirmative a conjecture by Lubiw and Sleumer [15, 17], that any biconnected outerplanar graph admits a Gabriel drawing. We then show that there exist biconnected outerplanat graphs that do not admit any convex β-drawing for 1 ≤ β ≤ 2. We also provide upper bounds on the maximum number of biconnected components sharing the same cut-vertex in β-drawable connected outerplanar graph. This last result is generalized to arbitrary connected planar graphs and is the first non-trivial characterization of connected β-drawable graphs. Finally, a weaker definition of proximity drawings is applied and we show that all connected outerplanar graphs are drawable under this definition.