On Constructing Regular Distance-Preserving Graphs

2014/05/07 by Abdol-Hossein Esfahanian, Ronald Nussbaum, Esfahanian, Abdol-Hossein +5
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Primary: 05C12 #Secondary: 05C07 #math.CO #msc:05C07 #msc:05C12

paper · pdf · doi:10.48550/arxiv.1405.1713

9 pages, 4 figures, 2 tables

arxiv created 2014/05/07 · arxiv updated 2014/05/08

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≤ 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.

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