# Rotation and jump distances between graphs

• Volume: 17, Issue: 2, page 285-300
• ISSN: 2083-5892

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## Abstract

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A graph H is obtained from a graph G by an edge rotation if G contains three distinct vertices u,v, and w such that uv ∈ E(G), uw ∉ E(G), and H = G-uv+uw. A graph H is obtained from a graph G by an edge jump if G contains four distinct vertices u,v,w, and x such that uv ∈ E(G), wx∉ E(G), and H = G-uv+wx. If a graph H is obtained from a graph G by a sequence of edge jumps, then G is said to be j-transformed into H. It is shown that for every two graphs G and H of the same order (at least 5) and same size, G can be j-transformed into H. For every two graphs G and H of the same order and same size, the jump distance ${d}_{j}\left(G,H\right)$ between G and H is defined as the minimum number of edge jumps required to j-transform G into H. The rotation distance ${d}_{r}\left(G,H\right)$ between two graphs G and H of the same order and same size is the minimum number of edge rotations needed to transform G into H. The jump and rotation distances of two graphs of the same order and same size are compared. For a set S of graphs of a fixed order at least 5 and fixed size, the jump distance graph ${D}_{j}\left(S\right)$ of S has S as its vertex set and where G₁ and G₂ in S are adjacent if and only if ${d}_{j}\left(G₁,G₂\right)=1$. A graph G is a jump distance graph if there exists a set S of graphs of the same order and same size with ${D}_{j}\left(S\right)=G$. Several graphs are shown to be jump distance graphs, including all complete graphs, trees, cycles, and cartesian products of jump distance graphs.

## How to cite

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Gary Chartrand, et al. "Rotation and jump distances between graphs." Discussiones Mathematicae Graph Theory 17.2 (1997): 285-300. <http://eudml.org/doc/270366>.

@article{GaryChartrand1997,
abstract = {A graph H is obtained from a graph G by an edge rotation if G contains three distinct vertices u,v, and w such that uv ∈ E(G), uw ∉ E(G), and H = G-uv+uw. A graph H is obtained from a graph G by an edge jump if G contains four distinct vertices u,v,w, and x such that uv ∈ E(G), wx∉ E(G), and H = G-uv+wx. If a graph H is obtained from a graph G by a sequence of edge jumps, then G is said to be j-transformed into H. It is shown that for every two graphs G and H of the same order (at least 5) and same size, G can be j-transformed into H. For every two graphs G and H of the same order and same size, the jump distance $d_j(G,H)$ between G and H is defined as the minimum number of edge jumps required to j-transform G into H. The rotation distance $d_r(G,H)$ between two graphs G and H of the same order and same size is the minimum number of edge rotations needed to transform G into H. The jump and rotation distances of two graphs of the same order and same size are compared. For a set S of graphs of a fixed order at least 5 and fixed size, the jump distance graph $D_j(S)$ of S has S as its vertex set and where G₁ and G₂ in S are adjacent if and only if $d_j(G₁,G₂) = 1$. A graph G is a jump distance graph if there exists a set S of graphs of the same order and same size with $D_j(S) = G$. Several graphs are shown to be jump distance graphs, including all complete graphs, trees, cycles, and cartesian products of jump distance graphs.},
author = {Gary Chartrand, Heather Gavlas, Héctor Hevia, Mark A. Johnson},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {edge rotation; rotation distance; edge jump; jump distance; jump distance graph},
language = {eng},
number = {2},
pages = {285-300},
title = {Rotation and jump distances between graphs},
url = {http://eudml.org/doc/270366},
volume = {17},
year = {1997},
}

TY - JOUR
AU - Gary Chartrand
AU - Heather Gavlas
AU - Héctor Hevia
AU - Mark A. Johnson
TI - Rotation and jump distances between graphs
JO - Discussiones Mathematicae Graph Theory
PY - 1997
VL - 17
IS - 2
SP - 285
EP - 300
AB - A graph H is obtained from a graph G by an edge rotation if G contains three distinct vertices u,v, and w such that uv ∈ E(G), uw ∉ E(G), and H = G-uv+uw. A graph H is obtained from a graph G by an edge jump if G contains four distinct vertices u,v,w, and x such that uv ∈ E(G), wx∉ E(G), and H = G-uv+wx. If a graph H is obtained from a graph G by a sequence of edge jumps, then G is said to be j-transformed into H. It is shown that for every two graphs G and H of the same order (at least 5) and same size, G can be j-transformed into H. For every two graphs G and H of the same order and same size, the jump distance $d_j(G,H)$ between G and H is defined as the minimum number of edge jumps required to j-transform G into H. The rotation distance $d_r(G,H)$ between two graphs G and H of the same order and same size is the minimum number of edge rotations needed to transform G into H. The jump and rotation distances of two graphs of the same order and same size are compared. For a set S of graphs of a fixed order at least 5 and fixed size, the jump distance graph $D_j(S)$ of S has S as its vertex set and where G₁ and G₂ in S are adjacent if and only if $d_j(G₁,G₂) = 1$. A graph G is a jump distance graph if there exists a set S of graphs of the same order and same size with $D_j(S) = G$. Several graphs are shown to be jump distance graphs, including all complete graphs, trees, cycles, and cartesian products of jump distance graphs.
LA - eng
KW - edge rotation; rotation distance; edge jump; jump distance; jump distance graph
UR - http://eudml.org/doc/270366
ER -

## References

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