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On feedback vertex sets and nonseparating independent sets in cubic graphs

Let G be an undirected connected graph with n nodes. A subset F of nodes of G is a feedback vertex set (fvs) if G − F is a forest and a subset J of nodes of G is a nonseparating independent set (nsis) if no two nodes of J are adjacent and G − J is connected. f(G), z(G) denote the cardinalities of a minimum fvs and a maximum nsis, respectively, of G. The equation f(G) = n/2 − z(G) + 1 and two new upper bounds on f(G) are derived for cubic graphs G.