Nested triangles graph
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.[1] 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.[2] 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.
Variants of the nested triangles graph have been used for many lower bound constructions in graph drawing, for instance on area of rectangular visibility representations,[3] area of drawings with right angle crossings[4] or relative area of planar versus nonplanar drawings.[5]
References
- ^ Dolev, Danny; Leighton, Tom; Trickey, Howard (1984), "Planar embedding of planar graphs" (PDF), Advances in Computing Research, 2: 147–161
- ^ Frati, Fabrizio; Patrignani, Maurizio (2008), "A note on minimum-area straight-line drawings of planar graphs", Graph Drawing: 15th International Symposium, GD 2007, Sydney, Australia, September 24-26, 2007, Revised Papers, Lecture Notes in Computer Science, vol. 4875, Berlin: Springer, pp. 339–344, doi:10.1007/978-3-540-77537-9_33, MR 2427831.
- ^ Fößmeier, Ulrich; Kant, Goos; Kaufmann, Michael (1997), "2-visibility drawings of planar graphs", in North, Stephen (ed.), Graph Drawing: Symposium on Graph Drawing, GD '96 Berkeley, California, USA, September 18–20, 1996, Proceedings, Lecture Notes in Computer Science, vol. 1190, pp. 155–168, doi:10.1007/3-540-62495-3_45.
- ^ Didimo, Walter; Liotta, Giuseppe (2013), "The crossing-angle resolution in graph drawing", in Pach, János (ed.), Thirty Essays on Geometric Graph Theory, Springer, pp. 167–184, doi:10.1007/978-1-4614-0110-0_10.
- ^ van Kreveld, Marc (2011), "The quality ratio of RAC drawings and planar drawings of planar graphs", in Brandes, Ulrik; Cornelsen, Sabine (eds.), Graph Drawing: 18th International Symposium, GD 2010, Konstanz, Germany, September 21-24, 2010, Revised Selected Papers, Lecture Notes in Computer Science, vol. 6502, pp. 371–376, doi:10.1007/978-3-642-18469-7_34.