2017/01/17 by Jason P. Smith, Smith, Jason P., Emad Zahedi +1
Computer Science · Mathematics · #05C12 #05C69 #Advanced Graph Theory Research #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Graph Labeling and Dimension Problems #Graph theory and applications #Social and Information Networks (cs.SI)
paper · pdf · doi:10.48550/arxiv.1701.06404
openalex publication_date 2017/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
A graph H is an isometric subgraph of G if dH(u,v)= dG(u,v), for every pair~u,v∈ V(H). A graph is distance preserving if it has an isometric subgraph of every possible order. A graph is sequentially distance preserving if its vertices can be ordered such that deleting the first i vertices results in an isometric subgraph, for all i≥1. We give an equivalent condition to sequentially distance preserving based upon simplicial orderings. Using this condition, we prove that if a graph does not contain any induced cycles of length~5 or greater, then it is sequentially distance preserving and thus distance preserving. Next we consider the distance preserving property on graphs with a cut vertex. Finally, we define a family of non-distance preserving graphs constructed from cycles.