Diameter Bounds for Planar Graphs

TL;DR

This paper establishes tight bounds on the diameter of connected planar graphs in relation to their inverse degree, introducing a new surgical method and providing precise upper bounds.

Contribution

It proves a tight upper bound on the diameter of planar graphs based on inverse degree and develops a novel surgical technique for analysis.

Findings

Diameter is at most 2.5 times the inverse degree in planar graphs.

Derived tight upper bounds for diameter based on vertices and edges.

Introduced a new surgical method for analyzing graph diameters.

Abstract

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 4V^2/3E on the diameter (for connected planar graphs), which are also tight.