On Constructing Regular Distance-Preserving Graphs

TL;DR

This paper introduces methods to construct regular graphs that preserve distances within subgraphs of all sizes, expanding understanding of distance-preserving structures in graph theory.

Contribution

The authors develop a construction technique for regular distance-preserving graphs of all sizes and degrees, and adapt the Havel-Hakimi algorithm for broader degree sequences.

Findings

Constructed regular distance-preserving graphs for all orders and degrees.

Modified the Havel-Hakimi algorithm to generate distance-preserving graphs for certain degree sequences.

Verified related conjectures computationally for small graph sizes.

Abstract

Let G be a simple, connected graph on n vertices. Let d_G(u,v) denote the distance between vertices u and v in G. A subgraph H of G is isometric if d_H(u,v)=d_G(u,v) for every u,v in V(H). We say that G is a distance-preserving graph if G contains at least one isometric subgraph of order k for every k, 1\le k\le n. In this paper we construct regular distance-preserving graphs of all possible orders and degrees of regularity. By modifying the Havel-Hakimi algorithm, we are able to construct distance preserving graphs for certain other degree sequences as well. We include a discussion of some related conjectures which we have computationally verified for small values of n.