Enumeration of three kinds of rooted maps on the Klein bottle (Q2454995)

\(N_{\widetilde g}\) denotes the nonorientable compact, closed 2-manifold with \(g\) crosscaps. A map \(M\) on \(N_{\widetilde g}\) is induced by the embedding of a connected graph in such a way that the residual components are homeomorphic to an open disk; \(M\) is \(\widetilde g\)-essential if, for each of its edges \(e\), \(M-e\) is ``either not a map, or a map which can not be embedded on the surface \(N_{\widetilde g}\)''. The maps being enumerated are rooted by a root-vertex, by a root-edge incident with that vertex, and by a root-face incident with that edge; a map is singular if it has a single face. In the 1960's W. T. Tutte and his students investigated methods of enumerating maps according to various criteria, cf.\ [\textit{W. T. Tutte}, Bull. Am. Math. Soc. 68, 500--504 (1962; Zbl 0109.41702)], which the reviewer applied to several classes of non-planar maps [\textit{W. G. Brown}, On the enumeration of non-planar maps, Mem. Am. Math. Soc. 65, 42 p. (1966; Zbl 0149.21201)]. From the authors' abstract and introduction: ``In this paper \(\widetilde2\)-essential rooted maps on the Klein bottle are counted, and an explicit expression with the size as a parameter is given. Further, the number of singular maps, and the maps with one vertex on the Klein bottle are derived. \dots At the same time a result of the corresponding problem in [\textit{R. Hao, J. Cai} and \textit{Y. Liu}, Korean J. Comput. Appl. Math. 9, No. 2, 451--463 (2002; Zbl 1006.05032)] is corrected.''