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

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) = maxJ{|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 γM(G), of G is equal to z(G). Then, according to this result, some new characterizations of the maximum genus γM(G) are obtained.