Non-embeddable extensions of embedded minors
R Hegde, R Thomas - arXiv preprint arXiv:1401.2973, 2014 - arxiv.org
R Hegde, R Thomas
arXiv preprint arXiv:1401.2973, 2014•arxiv.orgA graph G is weakly 4-connected if it is 3-connected, has at least five vertices, and for every
pair of sets (A, B) with union V (G) and intersection of size three such that no edge has one
end in AB and the other in BA, one of the induced subgraphs G [A], G [B] has at most four
edges. We describe a set of constructions that starting from a weakly 4-connected planar
graph G produce a finite list of non-planar weakly 4-connected graphs, each having a minor
isomorphic to G, such that every non-planar weakly 4-connected graph H that has a minor …
pair of sets (A, B) with union V (G) and intersection of size three such that no edge has one
end in AB and the other in BA, one of the induced subgraphs G [A], G [B] has at most four
edges. We describe a set of constructions that starting from a weakly 4-connected planar
graph G produce a finite list of non-planar weakly 4-connected graphs, each having a minor
isomorphic to G, such that every non-planar weakly 4-connected graph H that has a minor …
A graph G is weakly 4-connected if it is 3-connected, has at least five vertices, and for every pair of sets (A,B) with union V(G) and intersection of size three such that no edge has one end in A-B and the other in B-A, one of the induced subgraphs G[A], G[B] has at most four edges. We describe a set of constructions that starting from a weakly 4-connected planar graph G produce a finite list of non-planar weakly 4-connected graphs, each having a minor isomorphic to G, such that every non-planar weakly 4-connected graph H that has a minor isomorphic to G has a minor isomorphic to one of the graphs in the list. Our main result is more general and applies in particular to polyhedral embeddings in any surface.
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