Old and new generalizations of line graphs
Summary: Line graphs have been studied for over seventy years. \textit{H. Whitney} [Am. Math. J. 54, 150--168 (1932; Zbl 0003.32804)] showed that for connected graphs, edge-isomorphism implies isomorphism except for \(K_3\) and \(K_{1,3}\). 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.
scientific article; zbMATH DE number 2168702