The Forwarding Indices of Graphs -- a Survey

TL;DR

This survey reviews recent research on forwarding indices in graphs, which measure maximum load in network routing, highlighting known results, open problems, and conjectures in the field.

Contribution

It compiles and discusses existing results, open problems, and conjectures related to forwarding indices in graphs, providing a comprehensive overview of the topic.

Findings

Summarizes key results on vertex and edge forwarding indices.

Identifies open problems and conjectures in the field.

Highlights the importance of minimizing forwarding indices for network efficiency.

Abstract

A routing $R$ of a given connected graph $G$ of order $n$ is a collection of $n(n-1)$ simple paths connecting every ordered pair of vertices of $G$. The vertex-forwarding index $\xi(G,R)$ of $G$ with respect to $R$ is defined as the maximum number of paths in $R$ passing through any vertex of $G$. The vertex-forwarding index $\xi(G)$ of $G$ is defined as the minimum $\xi(G,R)$ over all routing $R$'s of $G$. Similarly, the edge-forwarding index $ \pi(G,R)$ of $G$ with respect to $R$ is the maximum number of paths in $R$ passing through any edge of $G$. The edge-forwarding index $\pi(G)$ of $G$ is the minimum $\pi(G,R)$ over all routing $R$'s of $G$. The vertex-forwarding index or the edge-forwarding index corresponds to the maximum load of the graph. Therefore, it is important to find routings minimizing these indices and thus has received much research attention in the past ten years and more. In this paper we survey some known results on these forwarding indices, further research problems and several conjectures.