Dominating Sets in Triangulations on Surfaces

TL;DR

This paper investigates bounds on the size of dominating sets in triangulations on surfaces, extending known results from planar graphs to more complex surfaces and degree constraints, with new bounds and cycle length results.

Contribution

It proves new bounds on dominating set sizes for triangulations on surfaces with degree constraints and establishes cycle length bounds for non-orientable surface triangulations.

Findings

Dominating set size at most n/6 + c for certain triangulations with degree constraints.

Existence of dominating sets of size at most n(1/6 + epsilon) + C on various surfaces.

Non-contractible cycles of length at most 2sqrt(n) in non-orientable surface triangulations.

Abstract

A dominating set D of a graph G is a set such that each vertex v of G is either in the set or adjacent to a vertex in the set. Matheson and Tarjan (1996) proved that any n-vertex plane triangulation has a dominating set of size at most n/3, and conjectured a bound of n/4 for n sufficiently large. King and Pelsmajer recently proved this for graphs with maximum degree at most 6. Plummer and Zha (2009) and Honjo, Kawarabayashi, and Nakamoto (2009) extended the n/3 bound to triangulations on surfaces. We prove two related results: (i) There is a constant c such that any n-vertex plane triangulation with maximum degree at most 6 has a dominating set of size at most n/6 + c. (ii) For any surface S, nonnegative t, and epsilon > 0, there exists C such that for any n-vertex triangulation on S with at most t vertices of degree other than 6, there is a dominating set of size at most n(1/6 + epsilon) + C. As part of the proof, we also show that any n-vertex triangulation of a non-orientable surface has a non-contractible cycle of length at most 2sqrt(n). Albertson and Hutchinson (1986) proved that for n-vertex triangulation of an orientable surface other than a sphere has a non-contractible cycle of length sqrt(2n), but no similar result was known for non-orientable surfaces.