Graph extensions, edit number and regular graphs

TL;DR

This paper studies how graphs can be extended by adding vertices with the same degree, introduces edit numbers to measure modifications, and presents an algorithm for constructing large regular graphs.

Contribution

It characterizes graphs with minimal edit numbers and provides an iterative method to build large connected regular graphs from smaller ones.

Findings

Graphs with zero edit number can be extended using regular graphs.

An iterative algorithm for constructing large regular graphs is proposed.

Characterization of graphs with minimal edit numbers is achieved.

Abstract

A graph G on n vertices is said to be extendable if G can be modified to form a new graph H on more than n vertices, while preserving the degrees of the vertices common to G and H. The added vertices all have the same degree and we define edit numbers to quantify the amount of modification needed to obtain the extended graph. Characterizing graphs with least possible edit numbers, we obtain that graphs with zero edit number can be extended using regular graphs. We also describe an iterative algorithm to construct connected regular graphs on arbitrarily large vertex sets, starting from the complete graph on a fixed set of vertices.