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By Kuhn D., Osthus D.

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As A separates G unless A = V (H), an obviously necessary condition for G being 3-connected, too, is |A| ≥ 3. ) The key observation for the generator theorem in this section is that |A| ≥ 3 is also sufficient for G being 3-connected. Lemma 7. Every mounted series parallel extension of some 3-connected graph at a set of at least 3 vertices is 3-connected. Proof. Let H, G, x, G, X, A be as in the beginning of this section, where we suppose that H is 3-connected and |A| ≥ 3. Then G, G are 2-connected, and s(G) = G − −x = H.

Let G be a graph of order n and maximum degree . If k ≥ 1 is an integer, then βk (G) ≥ Proof. If r = 1 + k n 1+ /k . 1002/jgt d(u) ≤ r 1+ /k

Clearly, not every mounted series parallel extension of some planar 3-connected graph H is planar again (K5 , say, is a series parallel extension of K4 ). How to characterize the planar ones in terms of H? What would be the planar analogue of Theorem 3? At first sight, a prospective starting point is to consider any series parallel extension G of some planar 3-connected graph H, because G is planar for free. But it might be impossible to add x to G such that it is adjacent to all vertices of degree 2 and such that the result remains planar, not regarding any additional connectivity constraints yet.

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A Note on Complete Subdivisions in Digraphs of Large Outdegree by Kuhn D., Osthus D.


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