This question asks about the forbidden minors of the class of graphs that can be formed by taking three clique sums of planar graphs and bounded treewidth graphs(The class is defined for some constant $t$), if there is a forbidden minors set for the class with all graphs have crossing number at most one. A reverse of the statement is known from Robertson and Seymours graph minor theory, that for any graph $H$ with crossing number at most one, there exists a constant, say $t$, such that any $H$ minor free graph can be written as a three clique sum of planar graphs and graphs with treewidth $\leq t$.

I want to ask a similar question for bounded genus graphs, that is, if $C$ is a class of graphs that can be expressed as three clique sums of bounded genus and bounded treewidth graphs, what can we say about the forbidden minors of the class? (It is a finite set since $C$ is minor closed right?)


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