On Proximity and other Distance Parameters in Planar Graphs

TL;DR

This paper improves bounds on proximity and other distance parameters in planar graphs and graphs with given connectivity, providing sharper inequalities and demonstrating their sharpness with example graphs.

Contribution

It significantly refines existing bounds on proximity-related parameters for maximal planar and highly connected graphs, introducing new bounds that depend on connectivity.

Findings

Bounds on the difference between radius and proximity are improved to about n/12, n/16, and n/20 for various classes.

Bounds on the difference between remoteness and proximity are improved to about n/(4κ) for κ-connected graphs.

Bounds on the difference between diameter and proximity are improved to about 3n/(4κ) for κ-connected graphs.

Abstract

Let $G$ be a connected graph. The average distance of a vertex $v$ of $G$ is the arithmetic mean of the distances from $v$ to all other vertices of $G$. The proximity and remoteness of $G$ are defined as the minimum and maximum, respectively, of the average distances of the vertices of $G$. It was shown by Aouchiche and Hansen [Proximity and remoteness in graphs: bounds and conjectures, Networks 58 no.\ 2 (2011)] that for a connected graph of order $n$, the difference between remoteness and proximity and the difference between radius and proximity are bounded from above by about $\frac{n}{4}$, and the difference between diameter and proximity is bounded from above by about $\frac{3}{4}n$. In this paper, we show that all three bounds can be improved significantly for maximal planar graphs, and for graphs of given connectivity. We show that in maximal planar graphs the above bound on the difference between radius and proximity can be improved to about $\frac{1}{12}n$, and further to about $\frac{1}{16}n$ and $\frac{1}{20}n$ if the graphs is, in addition, $4$-connected or $5$-connected, respectively. Similar improvements are shown for quadrangulations, and for maximal outerplanar graphs. We further show that the above bound on the difference between remoteness and proximity can be improved to about $\frac{1}{4\kappa}n$ if $G$ is $\kappa$-connected. Finally, we improve the bound on the difference between diameter and proximity to about $\frac{3}{4\kappa}n$ if $G$ is $\kappa$-connected. We present graphs that demonstrate that our bounds are either sharp, or sharp apart from an additive constant, even if restricted to planar graphs.