  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, 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.
