The Walk of Maximal Planar Graphs

Lee Zheng Han NUS High School, Singapore

S.-T. Yau High School Science Awarded Papers mathscidoc:1801.35033

Yau Science Award (Math), 2017.12
Polyhedral combinatorics has been a topic of interest in modern day’s computational geometry. The founding of Steinitz’s Theorem in 1922 revealed consequential relations between graph theory and polyhedral combinatorics. It allows us to better investigate on the topology of convex polyhedrons. In this paper, we proposed an algorithm that generates a unique sequence of points, using the vertices of a triangulated polyhedron, pre-determined by the selection of the starting 3 vertices in the sequence. Following that, we discover an interesting relation between the sequence and the volume of the polyhedron itself, in which we presented in the form of a sufficient condition. To further investigate which polyhedrons generate sequences that satisfy the sufficient condition, we study the problem in the context of graph theory, that is, the explorer walk (corresponding to the sequence of vertices) in maximal planar graphs (skeletons of triangulated convex polyhedrons). With that, we uncovered a family of maximal planar graphs, called the explorer graphs, which exhibits volumetric properties in the polyhedrons constructed from them, in regard to the explorer walk. In this paper, we also introduce generalized methods of constructing explorer graphs of higher order from explorer graphs of lower order, demonstrating the prevalence of explorer graphs. As the edges of a maximal planar graph is of great importance in tracing an explorer walk, we investigate on the line graph of maximal planar graphs, and re-establish a better definition of explorer graphs. Lastly, our paper covers the edge contraction of explorer graphs, which allows us to solve the volume of polyhedrons constructed from non-explorer graphs. For this, we presented a possible bound for the minimum number of edge contractions a non-explorer graph requires from an explorer graph. This will generalize the proposed method of finding volumes to any triangulated convex polyhedron.
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  title={The Walk of Maximal Planar Graphs},
  author={Lee Zheng Han},
  booktitle={Yau Science Award (Math)},
Lee Zheng Han. The Walk of Maximal Planar Graphs. 2017. In Yau Science Award (Math).
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