# Proximity drawings of outerplanar graphs (extended abstract)

## Abstract

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 *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 values 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 outerplanar 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 a *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.

## Keywords

Planar Graph Minimum Span Tree Delaunay Triangulation Adjacent Vertex Outerplanar Graph## References

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