Tech Report CS-96-21
Proximity Drawings of Outerplanar Graphs (Preliminary Version)
William Lenhart and Giuseppe Liotta
A proximity drawing of a graph is one in which pairs of adjacent vertices are 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 beta-drawings. These drawings include as special cases the well-known Gabriel drawings (when beta = 1), and relative neighborhood drawings (when beta = 2). We first show that all biconnected outerplanar graphs are beta-drawable for all values of beta such that beta is in the interval [1,2]. As a side effect, this result settles in the affirmative a conjecture by Lubiw and Sleumer, that any biconnected outerplanar graph admits a Gabriel drawing. We then show that there exist biconnected outerplanar graphs that do not admit any convex $\beta$-drawing for beta in the interval [1,2]. We also provide upper bounds on the maximum number of biconnected components sharing the same cut-vertex in a $\beta$-drawable connected outerplanar graph. This last result is generalized to arbitrary connected planar graphs and is the first non-trivial characterization of connected $\beta$-drawable graphs. Finally, a weaker definition of proximity drawings is applied and we show that all connected outerplanar graphs are drawable under this definition.