Polygonal tree

Polygonal tree --- многоугольное дерево.

A graph [math]\displaystyle{ G }[/math] is called a polygonal tree, if it consists of finitely many regular polygons (we assume any two distinct polygons be not coplanar) and has the following two properties:

(1) any two distinct polygons are disjoint or have exactly one edge in common (such an edge can be a common edge of several polygons),

(2) the diagram obtained by joining the centroids of the polygons to the mid-point of the common edge has no closed curve.

If all polygons of a polygonal tree [math]\displaystyle{ G }[/math] are the same, say [math]\displaystyle{ s }[/math]-gons, then [math]\displaystyle{ G }[/math] is called an [math]\displaystyle{ s }[/math]-gonal tree. 6-gonal tree is called hexagonal tree. Consider the diagram defined in condition 2. If we set the centroids and the mid-points of common edges of some polygons as "red" vertices and "green" vertices, respectively, and the straight line segments as edges of a bipartite graph, then this graph is a tree.