On Constructing Regular Distance-Preserving Graphs

Let G be a simple, connected graph on n vertices. Let dG(u, v) denote the distance between vertices u and v in G. A subgraph H of G is isometric if dH(u, v) = dG(u, v) for every u, v ∈ V (H). We say that G is a distancepreserving graph if G contains at least one isometric subgraph of order k for every k, 1 ≤ k ≤ 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.