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## Search Results

- Creator:
- Matos, Inês, Tejel, Javier, Garcia, Alfredo, Toth, Csaba D., Korman, Matias, Saumell, Maria, Silveira, Rodrigo I., and Hurtado, Ferran
- Description:
- We study biplane graphs drawn on a finite planar point set \(S\) in general position. This is the family of geometric graphs whose vertex set is \(S\) and can be decomposed into two plane graphs. We show that two maximal biplane graphs—in the sense that no edge can be added while staying biplane—may differ in the number of edges, and we provide an efficient algorithm for adding edges to a biplane graph to make it maximal. We also study extremal properties of maximal biplane graphs such as the maximum number of edges and the largest maximum connectivity over \(n\) -element point sets.
- Resource Type:
- Article
- Identifier:
- 0911-0119
- Campus Tesim:
- Northridge

- Creator:
- Hoffmann, Michael and Toth, Csaba D.
- Description:
- It is shown that every disconnected vertex-colored plane straight line graph with no isolated vertices can be augmented (by adding edges) into a connected plane straight line graph such that the new edges respect the coloring and the degree of every vertex increases by at most two. The upper bound for the increase of vertex degrees is best possible: there are input graphs that require the addition of two new edges incident to a vertex. The exclusion of isolated vertices is necessary: there are input graphs with isolated vertices that cannot be augmented to a connected vertex-colored plane straight line graph.
- Resource Type:
- Article
- Identifier:
- 0911-0119
- Campus Tesim:
- Northridge

- Creator:
- Silveira, Rodrigo I., Korman, Matias, Garcia, Alfredo, Hurtado, Ferran, Tejel, Javier, Matos, Inês, Toth, Csaba D., and Saumell, Maria
- Description:
- We study biplane graphs drawn on a finite point set S in the plane in general position. This is the family of geometric graphs whose vertex set is S and which can be decomposed into two plane graphs. We show that every sufficiently large point set admits a 5-connected biplane graph and that there are arbitrarily large point sets that do not admit any 6-connected biplane graph. Furthermore, we show that every plane graph (other than a wheel or a fan) can be augmented into a 4-connected biplane graph. However, there are arbitrarily large plane graphs that cannot be augmented to a 5-connected biplane graph by adding pairwise noncrossing edges.
- Resource Type:
- Article
- Identifier:
- 0911-0119
- Campus Tesim:
- Northridge