Sparse surface graphs

Abstract

This thesis is concerned with building certain classes of sparse and tight surface graphs via suitable sets of inductive moves from specified sets of irreducible surface graphs. In particular, we derive topological inductive constructions for (2,2)-tight surface graphs in the case of the sphere, the cylinder and the torus. We present theorems and results that determine the number of the irreducible (2,2)-tight sphere, cylinder and torus graphs. We also found all irreducible (2,2)-tight torus graphs using two independent approaches. One of these approaches is based on calculation by hand. We employed a mixture of brute force methods and theoretical insight into the structure of such graphs. The other approach is a computer assisted search. Consequently, we found that there are exactly 116 irreducible (2,2)-tight torus graphs. We also organise and present all irreducible (2,2)-tight torus graphs. This thesis also includes a geometric application of configurations to circular arcs, in the spirit of the Keobe-Andreev-Thurston circle packing theorem. We used the inductive construction of (2,2)-tight torus graphs to show that every (2,2) -tight torus graph is the contact graph of a collection of circular arcs in the flat torus.