Maximum genus and maximum nonseparating independent set of a 3-regular graph

Abstract

A set J ⊆ V is called a nonseparating independent set (nsis) of a connected graph G = ( V, E), if J is an independent set of G, i.e., E ∩ { uv | ∀ u, v ∈ J} = 0, and G − J is connected. We call z( G) = max J {| J|| J is an nsis of G} the nsis number of G. Let G be a 3-regular connected graph; we prove that the maximum genus, denoted by γ M( G), of G is equal to z( G). Then, according to this result, some new characterizations of the maximum genus γ M( G) are obtained.