GD 2013: Graph Drawing pp 155-160

# Strongly-Connected Outerplanar Graphs with Proper Touching Triangle Representations

• J. Joseph Fowler
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8242)

## Abstract

A proper touching triangle representation $$\mathcal{R}$$ of an n-vertex planar graph consists of a triangle divided into n non-overlapping triangles. A pair of triangles are considered to be adjacent if they share a partial side of positive length. Each triangle in $$\mathcal{R}$$ represents a vertex, while each pair of adjacent triangles represents an edge in the planar graph. We consider the problem of determining when a proper touching triangle representation exists for a strongly-connected outerplanar graph, which is biconnected and after the removal of all degree-2 vertices and outeredges, the resulting connected subgraph only has chord edges (w.r.t. the original graph). We show that such a graph has a proper representation if and only if the graph has at most two internal faces (i.e., faces with no outeredges).

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