Disk/Band Surfaces of Spatial Graphs

Abstract

In this paper, we show that, for any spatial embedding $\Gamma:G\to\mathbf{R}^3$ of a connected planar graph $G$, there exists a disk/band surface of $\Gamma(G)$ satisfying a certain linking condition. As an application of this result, it is proved that the homology class of $\Gamma(G)$ is determined only by the linking numbers of disjoint pairs in the set of boundary/outermost cycles with respect to a fixed planar embedding of $G$.