Genera and minors of multibranched surfaces

TL;DR

This paper introduces the concept of genus for multibranched surfaces, generalizes graph minors to these surfaces, and establishes inequalities providing upper bounds for their genus.

Contribution

It defines the genus of multibranched surfaces, extends the concept of minors to these surfaces, and investigates their properties and bounds.

Findings

Established inequalities for upper bounds of genus

Defined minors for multibranched surfaces

Analyzed properties of multibranched surface minors

Abstract

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.