Genera and minors of multibranched surfaces

We say that a 2-dimensional CW complex is a multibranched surface if we remove all points whose open neighborhoods are homeomorphic to the 2-dimensional Euclidean space R-2, then we obtain a 1-dimensional complex which is homeomorphic to a disjoint union of some S-1's. We define the genus of a multibranched surface X as the minimum number of genera of 3-dimensional manifold into which X can be embedded. We prove some inequalities which give upper bounds for the genus of a multibranched surface. A multibranched surface is a generalization of graphs. Therefore, we can define "minors" of multibranched surfaces analogously. We study various properties of the minors of multibranched surfaces. (C) 2017 Elsevier B.V. All rights reserved.