By Kuhn D., Osthus D.

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DH (z) = 3), for if A ∩ B consisted of its central vertex only then G would be a wheel again, contradicting our choice. Also note that if H ∼ = C3 ∗ K1 then G must be a proper supergraph of C4 ∗ K1 and thus contains a K4 as a proper subgraph. It remains to consider the case that H is a wheel C ∗ K1 for some > 3, and x is its central vertex. If |A| > 2, then we take c = d in B. The cycle H − x is the union of two edge disjoint c, d-paths P, Q where |V (P) ∩ A| ≥ 2. Let a1 , a2 , . . , a , ≥ 2, be the vertices from A on P where ai is closer to c than ai+1 .

Stracke and L. Volkmann, A new domination conception, J Graph Theory 17 (1993), 315–323. com). 20277 Abstract: To suppress a vertex v in a finite graph G means to delete it and add an edge from a to b if a, b are distinct nonadjacent vertices which formed the neighborhood of v. Let G − −x be the graph obtained from G − x by suppressing vertices of degree at most 2 as long as it is possible; this is proven to be well defined. Our main result states that every 3-connected graph G has a vertex x such that G − −x is 3-connected unless G is isomorphic to K 3,3 , K 2 × K 3 , or to a wheel K 1 ∗ C for some ≥ 3.

Let m = ln n . Let G be a k-detour subgraph of Qn , and (G) = be the maximum degree in G. Since G is connected, ≥ 2. Let u be a vertex of G. For each vertex v at distance m from u in Qn , we have m ≤ distG (u, v) ≤ m + k. 1002/jgt 58 JOURNAL OF GRAPH THEORY the number of walks in G of length j starting at u is at most n ≤ m k/2 m+2i <2 m+k j , we have . (6) i=0 Recall that m(m − 1) ≤ n for n ≥ 3 and therefore n nm 1 m−1 = 1− ... 1 − m m! n n ≥ nm m(m − 1) 1− m! 2n ≥ nm . 2m! Since n ≥ 21, we have m ≥ 3.

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

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