I am interested in work pertaining to graphs that have genus 1 i.e. toroidal graphs. Specifically, i am trying to find answers to the questions below.
Since toroidal graphs can be recognized in $P$ , what are some different known characterizations of toroidal graphs ?
There are more than thousand forbidden minors for toroidal graph class, and only four of them does not contain $K_{3,3}$ as a subdivision (This paper). Where can i find a bigger list of forbidden structures of toroidal graphs ?
Two disjoint copies of $K_5$'s are not toroidal. Is it true that if a graph $G$ have two vertex disjoint non-planar induced subgraphs, then $G$ is not a toridal ? If not, then what is special about disjoint copies of $K_5$'s ?