Diameter bounds for planar graphs

The inverse degree of a graph is the sum of the reciprocals of the degrees of its vertices. We prove that in any connected planar graph, the diameter is at most 5/2 times the inverse degree, and that this ratio is tight. To develop a crucial surgery method, we begin by proving the simpler related upper bounds ( 4 ( | V | − 1 ) − | E | ) / 3 and 4 | V | 2 / 3 | E | on the diameter (for connected planar graphs), which are also tight.