Contractible Edges and Removable Edges in 3-Connected Graphs

In this paper we prove that every spanning tree of a 3-connected 3-regular graph, except for K4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_4$$\end{document} and K2□K3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_2\, \Box \,K_3$$\end{document}, contains at least two contractible edges, and that every spanning tree of a minimally 3-connected graph, except for the wheel graphs, contains at least one contractible edge. Moreover, we show that every maximum matching of a 3-connected graph of order at least 6 that does not contain the maximal semiwheel graphs avoids at least four removable edges; every maximum matching of a 3-connected graph avoids at least two removable edges.