2 research outputs found
Non-Embeddable Extensions of Embedded Minors
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.Comment: 30 pages, 3 figure
Non-planar extensions of subdivisions of planar graphs
Almost -connectivity is a weakening of -connectivity which allows for
vertices of degree three. In this paper we prove the following theorem. Let
be an almost -connected triangle-free planar graph, and let be an almost
-connected non-planar graph such that has a subgraph isomorphic to a
subdivision of . Then there exists a graph such that is isomorphic
to a minor of , and either
(i) for some vertices such that no facial cycle of
contains both and , or
(ii) for some distinct vertices such that appear on some facial cycle of in the
order listed.
This is a lemma to be used in other papers. In fact, we prove a more general
theorem, where we relax the connectivity assumptions, do not assume that is
planar, and consider subdivisions rather than minors. Instead of face
boundaries we work with a collection of cycles that cover every edge twice and
have pairwise connected intersection. Finally, we prove a version of this
result that applies when is planar for some set of size at most , but is non-planar for every set
of size at most .Comment: This version fixes an error in the published paper. The error was
kindly pointed out to us by Katherine Naismith. Changes from the published
version are indicated in red. 57 pages, 8 figure