31. This is hard to search for since you just get results for " planar graph ". 32. For a planar graph , the nullity is equal to the number of meshes in the graph. 33. They generalize to matroids the notions of planar graph duality and of dual spaces in linear algebra. 34. However, W . T . Tutte showed that all 4-connected planar graphs are hamiltonian. 35. Apollonian networks are the maximal planar graphs formed by repeatedly splitting triangular faces into triples of smaller triangles. 36. F�ry's theorem states that any planar graph may be represented as a planar straight line graph. 37. However, unlike treewidth, the branchwidth of planar graphs may be computed exactly, in polynomial time. 38. This in contrast to treewidth for which the complexity on planar graphs is a well known open problem. 39. In particular, there exists a planar graph without 4-cycles that cannot be 3-colored. 40. Every maximal planar graph with five or more vertices has vertex connectivity 3, 4, or 5.
