In this paper several results are proved: (1) Deciding whether a given planar graph G with maximum degree 3 has a spanning tree T, where deg(x, T) = 1 or 3 for each node x, is NP-Complete. (2) For each proper subset S of the positive integers + with 1 ϵ S and |S|⩾2, deciding whether a planar graph G = (V, E) has a spanning tree T such that deg(x, T) ϵ S, for all x ϵ V, is NP-Complete. (3) For each non-empty proper subset S of +, deciding whether, given a planar graph G = (V, E) and integer N with 1⩽N<|V|, there is a spanning tree T for G such that T has at least [at most] N nodes x with deg(x, T) ϵ S is NP-complete. Also, as corollaries of (1), we show that for a planar graph G, with n nodes and maximum degree 3, it is NP-complete to decide (4) if there is a spanning tree with at least [exactly] n/2 + 1 leaves and (5) if G has a connected dominating set with cardinality ⩽n/2–1.
