Nested triangles graph

A nested triangles graph with 18 vertices

Grid drawing of a nested triangles graph. This layout may be used to draw any such graph within a bounding box of dimensions (n/3 + O(1)) × (n/2 + O(1)). This doesn't improve the bound of Frati & Patrignani (2008), because their bound applies more generally to maximal graphs formed by adding diagonals to the quadrilaterals of this graph, and not every maximal graph can be drawn in this way.

In graph theory and graph drawing, a nested triangles graph with n vertices is a planar graph formed from a sequence of n/3 triangles, by connecting pairs of corresponding vertices on consecutive triangles in the sequence. It can also be formed geometrically, by gluing together n/3 − 1 triangular prisms on their triangular faces.

The nested triangles graph was named by Dolev, Leighton & Trickey (1984), who used it to show that drawing an n-vertex planar graph in the integer lattice (with straight line-segment edges) may require a bounding box of size at least n/3 × n/3. In such a drawing, no matter which face of the graph is chosen to be the outer face, some subsequence of at least n/6 of the triangles must be drawn nested within each other, and within this part of the drawing each triangle must use two rows and two columns more than the next inner triangle. Frati & Patrignani (2008) showed that this graph, and any graph formed by adding diagonals to its quadrilaterals, can be drawn within a box of dimensions n/3 × 2n/3. Although the nested triangles graph itself can be drawn in smaller area, closing the gap between this 2n²/9 upper bound and the n²/9 lower bound on drawing area for completions of the nested triangle graph remains an open problem. If the outer face is not allowed to be chosen as part of the drawing algorithm, but is specified as part of the input, the same argument shows that a bounding box of size 2n/3 × 2n/3 is necessary, and a drawing with these dimensions exists.