41. Strangulated graphs are graphs that can be formed by clique-sums of chordal graphs and maximal planar graphs . 42. Flat embeddings are automatically linkless, but not vice versa . and include the planar graphs and apex graphs. 43. If a planar graph is embedded on a sphere, its face cycles clearly satisfy Lefschetz's property. 44. The Circle packing theorem states that the intersection graphs of non-crossing circles are exactly the planar graphs . 45. Frederickson proposed another faster algorithm for single source shortest paths by implementing separator theorem in planar graphs in 1986. 46. Therefore, by Steinitz's theorem, it is a 3-vertex-connected simple planar graph . 47. In planar graphs , the set of bounded cycles of an embedding of the graph forms a cycle basis. 48. The minimum weight cycle basis of a planar graph corresponds to the Gomory Hu tree of the dual graph. 49. The method of for combining circle packings and quadtrees to achieve bounded slope number for planar graphs with bounded degree 50. However, a three-dimensional analogue of the planar graphs is provided by the topologically linked with each other.
