  In this paper, we consider the problem of representing graphs by triangles
whose sides touch. As a simple necessary condition, we show that pairs of
vertices must have a small common neighborhood. On the positive side, we
present linear time algorithms for creating touching triangle representations
for outerplanar graphs, square grid graphs, and hexagonal grid graphs. We note
that this class of graphs is not closed under minors, making characterization
difficult. However, we present a complete characterization of the subclass of
biconnected graphs that can be represented as triangulations of some polygon.
