On Touching Triangle Graphs

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.

💡 Research Summary

The paper introduces and studies a geometric representation of graphs in which each vertex is associated with a triangle in the plane and two vertices are adjacent precisely when the corresponding triangles share an entire side. This “touching‑triangle” model lies between classic contact graphs (where any part of a shape may touch) and more restrictive intersection representations, and it raises both combinatorial and algorithmic questions.

The authors first establish a simple necessary condition: for any edge (u,v) the common neighbourhood N(u)∩N(v) must be bounded by a constant independent of the graph size. The proof exploits the fact that a single triangle side can be adjacent to only a limited number of other triangle sides without overlap; consequently dense substructures such as large cliques cannot appear in a touching‑triangle graph. This condition already rules out many natural families (e.g., graphs containing K5 as a subgraph) and provides a quick test for impossibility.

Next, the paper presents linear‑time constructive algorithms for three important graph families.

1. Outerplanar graphs – Using a block‑decomposition and a depth‑first traversal, each biconnected component is embedded as a set of triangles whose sides meet exactly as required. The algorithm inserts triangles one by one, always attaching a new triangle to an existing side that corresponds to the edge being realized. Because outerplanar graphs have a planar embedding with all vertices on the outer face, the side‑attachment process never creates conflicts, yielding an O(n) algorithm.

2. Square‑grid graphs – The regular grid is processed cell by cell. Each unit square is split into two right‑angled triangles, and adjacent squares are aligned so that the shared grid edge becomes a shared triangle side. The construction proceeds in a raster order, guaranteeing that each grid edge is realized exactly once. The method scales linearly with the number of vertices.

3. Hexagonal‑grid graphs – Each hexagon is first triangulated into three equilateral triangles. By carefully orienting these triangles across the whole lattice, the authors ensure that every edge of the hexagonal grid corresponds to a common side of two neighboring triangles. Again the algorithm visits each hexagon once, achieving linear time.

These three algorithms demonstrate that large, highly regular families admit efficient touching‑triangle representations despite the restrictive side‑sharing rule.

A striking negative result follows: the class of touching‑triangle graphs is not closed under graph minors. The authors exhibit a small graph that admits a touching‑triangle representation while a minor obtained by edge contraction (or deletion) does not. For example, a K4 minus one edge can be realized, but contracting the missing edge yields K4, which cannot be represented because its four vertices would require each triangle to share sides with three others, exceeding the geometric bound from the necessary condition. This non‑closure implies that classic minor‑based characterizations (e.g., Kuratowski’s theorem for planarity) are unlikely to exist for this model, making a full structural description challenging.

Finally, the paper gives a complete characterization of biconnected graphs that can be represented as touching‑triangle graphs. The authors define “polygon triangulation graphs”: graphs that arise as the dual of a triangulation of a simple polygon, where each triangle of the triangulation becomes a vertex and each shared edge becomes an adjacency. They prove that a biconnected graph admits a touching‑triangle representation if and only if it is isomorphic to such a polygon triangulation graph. The proof proceeds in two directions: (i) given a polygon triangulation, construct a touching‑triangle layout by placing each triangle exactly where it appears in the polygon; (ii) given a touching‑triangle representation of a biconnected graph, recover a polygon whose triangulation yields the original graph by tracing the outer boundary formed by the outer sides of the triangles. This bijection supplies a polynomial‑time recognition algorithm for the biconnected case and clarifies the exact structural nature of the representable subclass.

In summary, the paper makes several contributions: it identifies a fundamental combinatorial restriction (bounded common neighbourhood), provides linear‑time algorithms for outerplanar, square‑grid, and hexagonal‑grid graphs, demonstrates that the class is not minor‑closed, and delivers a full characterization for biconnected graphs via polygon triangulations. These results open new avenues for exploring geometric graph representations where side‑to‑side contact is the sole mode of adjacency, and they suggest that further work will need to address the gap between the tractable families presented and the broader, more complex graphs that fall outside the touching‑triangle paradigm.