Reference request: Toroidal graph

Question

I have asked a similar question here but not sure if it has reached the right community.

I need reference to learn about graphs that have genus 1 i.e. toroidal graphs. Specifically, i am trying to find answers to the questions below.

1. Since toroidal graphs can be recognized in polynomial time, what are different known characterizations of toroidal graphs ?

2. It is known that 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 ?

3. 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 ?

Answer 1

For (2), take a look at this paper:

Myrvold, Wendy; Woodcock, Jennifer, A large set of torus obstructions and how they were discovered, Electron. J. Comb. 25, No. 1, Research Paper P1.16, 17 p. (2018). ZBL1380.05134.

Abstract

We outline the progress made so far on the search for the complete set of torus obstructions and also consider practical algorithms for torus embedding and their implementations. We present the set of obstructions that are known to-date and give a brief history of how these graphs were found. We also describe a nice algorithm for embedding graphs on the torus which we used to verify previous results and add to the set of torus obstructions. Although it is still exponential in the order of the graph, the algorithm presented here is relatively simple to describe and implement and fast-in-practice for small graphs. It parallels the popular quadratic planar embedding algorithm of Demoucron, Malgrange, and Pertuiset.

In section 6 of the paper, the authors provide links to a database of torus obstructions:

6 The Torus Obstructions

The torus obstruction described in the paper are available from:

http://www.combinatorics.org/ojs/index.php/eljc/article/view/v25i1p16/html

and

http://webhome.cs.uvic.ca/~wendym/torus/torus_obstructions.html