Strongly-Connected Outerplanar Graphs with Proper Touching Triangle Representations

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.