Partial triangulations of surfaces with girth constraints

TL;DR

This paper investigates finite classes of partial triangulations of surfaces with specific girth constraints, characterizing sparse graphs and establishing finiteness results for contraction-minimal graphs under certain tightness conditions.

Contribution

It extends finiteness results to cellular partial triangulations with girth inequalities and characterizes M-embedded sparse graphs via higher genus girth inequalities.

Findings

Finiteness of minimal triangulations for certain surface classes.

Characterization of M-embedded sparse graphs through girth inequalities.

Finiteness of contraction-minimal (3,6)-tight and (3,3)-tight graphs.

Abstract

Barnette and Edelson have shown that there are finitely many minimal triangulations of a connected compact 2-manifold M. Similar finiteness results are obtained for cellular partial triangulations that satisfy various girth inequality constraints for embedded cycles. A characterisation of various M-embedded sparse graphs is given in terms of the satisfaction of higher genus girth inequalities. With this it is shown that there are finitely many contraction-minimal M-embedded graphs that are (3,6)-tight or (3,3)-tight.