<p>Regarding (3), yes, 
if a graph $M$ has two vertex disjoint non-planar induced subgraphs $G$ and $H$, then $G\cup H$ (and hence $M$) is not toroidal.</p>

<p>I don't know a reference but here's a proof sketch.</p>

<p>Thinking of the torus as $T=S^1\times S^1$, if a non-planar $G$ is embedded without crossing on $T$ then its edges must meet every $S^1\times\{a\}$ and $\{a\}\times S^1$, or else we cut such a circle out and realize $G$ as planar. By tracing where they so meet we obtain a subgraph of $G$ realized as homotopic to some
$$\left(\{a\}\times S^1\right)\cup \left(S^1\times\{b\}\right)$$
(so two circles meet in $(a,b)$). Similarly for the second graph $H$ and say 
$$\left(\{c\}\times S^1\right)\cup \left(S^1\times\{d\}\right).$$
But then $G$ and $H$ are not disjoint (they share a vertex or have crossing edges) as the embedded graphs meet in $(a,d)$ and $(c,b)$.</p>

CC BY-SA 3.0
2017-11-02T08:02:36.963Z
2017-11-03T21:13:06.460Z
2017-11-03T21:13:06.460Z
Bjørn Kjos-Hanssen
<p>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.</p>

<ol>
<li><p>Since toroidal graphs can be recognized in $P$ , what are some different known characterizations of toroidal graphs ? </p></li>
<li><p>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 (<a href="http://www.sciencedirect.com/science/article/pii/S0012365X07001008" rel="nofollow noreferrer">This paper</a>). Where can i find a bigger list of forbidden structures of toroidal graphs ?</p></li>
<li><p>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 ?</p></li>
</ol>

<reference-request><graph-theory><graph-minor><topological-graph-theory>
Properties of toroidal graph
Dibyayan
# Properties of toroidal graph

(Asked by: username_0 on 2017-11-01)

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.

1. Since toroidal graphs can be recognized in $P$ , what are some different known characterizations of toroidal graphs ?
2. 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](http://www.sciencedirect.com/science/article/pii/S0012365X07001008)). 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 ?

**Tags:** reference-request, graph-theory, graph-minor, topological-graph-theory


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## 1 answers

### Answer by username_1 on 2017-11-02. Score: 2

Regarding (3), yes,
if a graph $M$ has two vertex disjoint non-planar induced subgraphs $G$ and $H$, then $G\cup H$ (and hence $M$) is not toroidal.

I don't know a reference but here's a proof sketch.

Thinking of the torus as $T=S^1\times S^1$, if a non-planar $G$ is embedded without crossing on $T$ then its edges must meet every $S^1\times\{a\}$ and $\{a\}\times S^1$, or else we cut such a circle out and realize $G$ as planar. By tracing where they so meet we obtain a subgraph of $G$ realized as homotopic to some
$$\left(\{a\}\times S^1\right)\cup \left(S^1\times\{b\}\right)$$
(so two circles meet in $(a,b)$). Similarly for the second graph $H$ and say
$$\left(\{c\}\times S^1\right)\cup \left(S^1\times\{d\}\right).$$
But then $G$ and $H$ are not disjoint (they share a vertex or have crossing edges) as the embedded graphs meet in $(a,d)$ and $(c,b)$.


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