Topological inductive constructions for tight surface graphs

TL;DR

This paper explores the properties of sparse and tight surface graphs, providing topological inductive constructions for specific cases and identifying all irreducible base graphs on the torus, with applications to contact graphs of circular arcs.

Contribution

It introduces new topological inductive constructions for (2, 2)-tight surface graphs and catalogs all irreducible base graphs on the torus, advancing understanding of their structure.

Findings

Derived inductive constructions for (2, 2)-tight surface graphs on various surfaces.

Identified all 116 irreducible base graphs on the torus.

Applied results to contact graphs of circular arc configurations.

Abstract

We investigate properties of sparse and tight surface graphs. In particular we derive topological inductive constructions for $(2, 2)$-tight surface graphs in the case of the sphere, the plane, the twice punctured sphere and the torus. In the case of the torus we identify all 116 irreducible base graphs and provide a geometric application involving contact graphs of configurations of circular arcs.