Face sizes and the connectivity of the dual

TL;DR

This paper establishes tight bounds on face sizes necessary for certain cuts in simple duals of c-connected maps, determining minimal genus conditions for these configurations.

Contribution

It provides the first tight bounds on face sizes for 1- and 2-cuts in simple duals of c-connected maps and identifies minimal genus conditions for their existence.

Findings

Tight lower bounds on face sizes for 1- and 2-cuts.

Minimal genus for c-connected maps with simple duals having 2-cuts.

Bounds for genus with 1-cuts in simple duals.

Abstract

For each $c\ge 1$ we prove tight lower bounds on face sizes that must be present to allow $1$- or $2$-cuts in simple duals of $c$-connected maps. Using these bounds, we determine the smallest genus on which a $c$-connected map can have a simple dual with a $2$-cut and give lower and some upper bounds for the smallest genus on which a $c$-connected map can have a simple dual with a $1$-cut.