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 between G and H is defined as the minimum number of edge jumps required to j-transform G into H. The rotation distance 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 of S has S as its vertex set and where G₁ and G₂ in S are adjacent if and only if . A graph G is a jump distance graph if there exists a set S of graphs of the same order and same size with . Several graphs are shown to be jump distance graphs, including all complete graphs, trees, cycles, and cartesian products of jump distance graphs.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1056,
author = {Gary Chartrand and Heather Gavlas and H\'ector Hevia and Mark A. Johnson},
title = {Rotation and jump distances between graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {17},
year = {1997},
pages = {285-300},
zbl = {0902.05022},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1056}
}
Gary Chartrand; Heather Gavlas; Héctor Hevia; Mark A. Johnson. Rotation and jump distances between graphs. Discussiones Mathematicae Graph Theory, Tome 17 (1997) pp. 285-300. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1056/
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