On Distance Preserving and Sequentially Distance Preserving Graphs

Jason P Smith, Emad Zahedi

Research output: Working paper

Abstract

A graph $H$ is an \emph{isometric} subgraph of $G$ if $d_H(u,v)= d_G(u,v)$, for every pair~$u,v\in V(H)$. A graph is \emph{distance preserving} if it has an isometric subgraph of every possible order. A graph is \emph{sequentially distance preserving} if its vertices can be ordered such that deleting the first $i$ vertices results in an isometric subgraph, for all $i\ge1$. 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.
Original languageEnglish
PublisherArXiv
Publication statusPublished - 17 Jan 2017

Fingerprint

Graph in graph theory
Isometric
Subgraph
Cut Vertex
Cycle

Keywords

  • cs.DM
  • cs.SI
  • math.CO

Cite this

On Distance Preserving and Sequentially Distance Preserving Graphs. / Smith, Jason P; Zahedi, Emad.

ArXiv, 2017.

Research output: Working paper

@techreport{d9b4d36a3fb44499ba6327cf6ec5467b,
title = "On Distance Preserving and Sequentially Distance Preserving Graphs",
abstract = "A graph $H$ is an \emph{isometric} subgraph of $G$ if $d_H(u,v)= d_G(u,v)$, for every pair~$u,v\in V(H)$. A graph is \emph{distance preserving} if it has an isometric subgraph of every possible order. A graph is \emph{sequentially distance preserving} if its vertices can be ordered such that deleting the first $i$ vertices results in an isometric subgraph, for all $i\ge1$. 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.",
keywords = "cs.DM, cs.SI, math.CO",
author = "Smith, {Jason P} and Emad Zahedi",
year = "2017",
month = "1",
day = "17",
language = "English",
publisher = "ArXiv",
type = "WorkingPaper",
institution = "ArXiv",

}

TY - UNPB

T1 - On Distance Preserving and Sequentially Distance Preserving Graphs

AU - Smith, Jason P

AU - Zahedi, Emad

PY - 2017/1/17

Y1 - 2017/1/17

N2 - A graph $H$ is an \emph{isometric} subgraph of $G$ if $d_H(u,v)= d_G(u,v)$, for every pair~$u,v\in V(H)$. A graph is \emph{distance preserving} if it has an isometric subgraph of every possible order. A graph is \emph{sequentially distance preserving} if its vertices can be ordered such that deleting the first $i$ vertices results in an isometric subgraph, for all $i\ge1$. 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.

AB - A graph $H$ is an \emph{isometric} subgraph of $G$ if $d_H(u,v)= d_G(u,v)$, for every pair~$u,v\in V(H)$. A graph is \emph{distance preserving} if it has an isometric subgraph of every possible order. A graph is \emph{sequentially distance preserving} if its vertices can be ordered such that deleting the first $i$ vertices results in an isometric subgraph, for all $i\ge1$. 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.

KW - cs.DM

KW - cs.SI

KW - math.CO

M3 - Working paper

BT - On Distance Preserving and Sequentially Distance Preserving Graphs

PB - ArXiv

ER -