Dominating sets in triangulations on surfaces

Abstract

A dominating set D ⊆ V ( G ) of a graph G is a set such that each vertex v ∈ V ( 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 1 such that any n -vertex plane triangulation with maximum degree at most 6 has a dominating set of size at most n /6 + c 1 . (ii) For any surface S , t ≥ 0, and ε > 0, there exists c 2 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 + ε ) + c 2 . 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 2√ 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 √(2 n ), but no similar result was known for non-orientable surfaces.