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International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 29, Pages 1509-1521
http://dx.doi.org/10.1155/S0161171204310094

Old and new generalizations of line graphs

Department of Computer Science, Ball State University, Muncie 47306, IN, USA

Received 8 October 2003

Copyright © 2004 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Line graphs have been studied for over seventy years. In 1932, H. Whitney showed that for connected graphs, edge-isomorphism implies isomorphism except for K3 and K1,3. The line graph transformation is one of the most widely studied of all graph transformations. In its long history, the concept has been rediscovered several times, with different names such as derived graph, interchange graph, and edge-to-vertex dual. Line graphs can also be considered as intersection graphs. Several variations and generalizations of line graphs have been proposed and studied. These include the concepts of total graphs, path graphs, and others. In this brief survey we describe these and some more recent generalizations and extensions including super line graphs and triangle graphs.