## Abstract A graph __H__ is __collapsible__ if for every subset X β __V(H), H__ has a spanning connected subgraph whose set of oddβdegree vertices is X. In any graph __G__ there is a unique collection of maximal collapsible subgraphs, and when all of them are contracted, the resulting contraction
On the existence of graphs of diameter two and defect two
β Scribed by J. Conde; J. Gimbert
- Book ID
- 108114085
- Publisher
- Elsevier Science
- Year
- 2009
- Tongue
- English
- Weight
- 459 KB
- Volume
- 309
- Category
- Article
- ISSN
- 0012-365X
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We survey what is known on geodetic graphs of diameter two and discuss the implications of a new strong necessary condition for the existence of such graphs.
A maximal planar graph is a simple planar graph in which every face is a triangle. We show here that such graphs with maximum degree A and diameter two have no more than :A + 1 vertices. We also show that there exist maximal planar graphs with diameter two and exactly LiA + 1 J vertices.
Skoviera, M., The maximum genus of graphs of diameter two, Discrete Mathematics 87 (1991) 175-180. Let G be a (finite) graph of diameter two. We prove that if G is loopless then it is upper embeddable, i.e. the maximum genus y,&G) equals [fi(G)/Z], where /3(G) = IF(G)1 -IV(G)1 + 1 is the Betti numbe