Trivalent 2-Arc Transitive Graphs of Type \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${G^1_2}$$\end{document} are Near Polygonal

A connected graph Σ of girth at least four is called a near n-gonal graph with respect to E, where n ≥  4 is an integer, if E is a set of n-cycles of Σ such that every path of length two is contained in a unique member of E. It is well known that connected trivalent symmetric graphs can be classified into seven types. In this note we prove that every connected trivalent G-symmetric graph \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Sigma \neq K_4}$$\end{document} of type \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${G^1_2}$$\end{document} is a near polygonal graph with respect to two G-orbits on cycles of Σ. Moreover, we give an algorithm for constructing the unique cycle in each of these G-orbits containing a given path of length two.