Rooted HIST property on planar triangulations

A spanning tree with no vertices of degree two of a graph is called a homeomorphically irreducible spanning tree (or HIST) of the graph. It has been proved that every planar triangulation G with at least four vertices has a HIST H [1]. However, the previous result asserts nothing whether the degree of a fixed vertex v of G is at least three or not in H. In this paper, we prove that if a planar triangulation G has 2n (n >= 2) vertices, then, for any vertex v, G has a HIST H such that the degree of v is at least three in H. We call such a spanning tree a rooted HIST of G with root v.