On Domination Number of Triangulated Discs

Abstract

Let G be a 3-connected triangulated disc such that the boundary cycle C of the outer face is an induced cycle of G and G-C is a tree. In this paper we prove that $\gamma(G) \leq \frac{n+2}{4}$, which gives a partial solution for the conjecture that the same inequality holds for any 3-conneced triangulated disc. We also show related conjectures.