Discussiones Mathematicae Graph Theory  17(2) (1997)  285-300
doi: 10.7151/dmgt.1056

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Gary Chartrand

Department of Mathematics and Statistics
Western Michigan University, Kalamazoo, MI 49008, USA

Heather Gavlas

Smiths Industries, Defense Systems North America
Grand Rapids, MI 49518-3469, USA

Héctor Hevia

Escuela de Ingenieria Comercial, Universidad Adolfo Ibanez
Balmaceda 1625, Vina del Mar, CHILE

Mark A. Johnson

Pharmacia & Upjohn, 7247-267-133
301 Henrietta Street, Kalamazoo, MI 49007, USA


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 dj(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 dr(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 Dj(S) of S has S as its vertex set and where G1 and G2 in S are adjacent if and only if dj(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 Dj(S) = G. Several graphs are shown to be jump distance graphs, including all complete graphs, trees, cycles, and cartesian products of jump distance graphs.

Keywords: edge rotation, rotation distance, edge jump, jump distance, jump distance graph.

1991 Mathematics Subject Classification: Primary: 05C12, Secondary: 05C75.


[1] V. Balá, J. Koa, V. Kvasnika and M. Sekanina, A metric for graphs, asopis Pst. Mat. 111 (1986) 431-433.
[2] G. Benadé, W. Goddard, T.A. McKee and P.A. Winter, On distances between isomorphism classes of graphs, Math. Bohemica 116 (1991) 160-169.
[3] G. Chartrand, W. Goddard, M.A. Henning, L. Lesniak, H.C. Swart and C.E. Wall, Which graphs are distance graphs? Ars Combin. 29A (1990) 225-232.
[4] G. Chartrand, F. Saba and H-B Zou, Edge rotations and distance between graphs, asopis Pst. Mat. 110 (1985) 87-91.
[5] R.J. Faudree, R.H. Schelp, L. Lesniak, A. Gyárfás and J. Lehel, On the rotation distance of graphs, Discrete Math. 126 (1994) 121-135, doi: 10.1016/0012-365X(94)90258-5.
[6] E.B. Jarrett, Edge rotation and edge slide distance graphs, Computers and Mathematics with Applications, (to appear).
[7] C. Jochum, J. Gasteiger and I. Ugi, The principle of minimum chemical distance, Angewandte Chemie International 19 (1980) 495-505, doi: 10.1002/anie.198004953.
[8] M. Johnson, Relating metrics, lines and variables defined on graphs to problems in medicinal chemistry, in: Graph Theory With Applications to Algorithms and Computer Science, Y. Alavi, G. Chartrand, L. Lesniak, D.R. Lick, and C.E. Wall, eds., (Wiley, New York, 1985) 457-470.
[9] V. Kvasnika and J. Pospichal, Two metrics for a graph-theoretic model of organic chemistry, J. Math. Chem. 3 (1989) 161-191, doi: 10.1007/BF01166047.