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

A set J subset of or equal to V is called a nonseparating independent set (nsis) of a connected graph G = (V,E), ii. J is an independent set of G, i.e., E boolean AND {u upsilon/For All u, upsilon is an element of 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 gamma(M)(G), of G is equal to z(G). Then, according to this result, some new characterizations of the maximum genus gamma(M)(G) are obtained.