Homeomorphically Irreducible Spanning Trees In Locally Connected Graphs

A spanning tree without vertices of degree 2 is called a homeomorphically irreducible spanning tree (HIST). A. Hill conjectured that every triangulation of the plane other than K3 contains a HIST. J. Malkevitch extended this conjecture to a near-triangulation of the plane (a 2-connected plane graph with all but at most one faces are triangles). Albertson, Berman, Hutchinson, and Thomassen confirmed the conjecture. Moreover, they asked whether every triangulation of a surface contains a HIST. In this paper, we show that every connected and locally connected graph with more than 3 vertices contains a HIST. Consequently, every triangulation of a surface contains a HIST.