Nakamoto, Oda and Ota 3-trees with few vertices of degree 3 in circuit graphs, Discrete Math. 309 (2009), 666–672 proved that every 3-connected planar graph with n vertices has a spanning 3-tree with at most max0, n−7 3 vertices of degree 3. This result was proved for a wider class of graphs, called the circuit graphs, and the result for circuit graphs is best possible for the pseudoradial graph of triangulations (PRTs). In this paper, we prove that only PRTs attain the bound, and consequently, we improve the bound for 3-connected planar graphs. Our result is a successor of our previous result on the size of 2-connected spanning subgraphs of circuit graphs and 3-connected planar graphs, but we extend our arguments on the two topics to 3-connected graphs on the projective plane.
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