Line Graphs and Beyond

The line graph operation, in which the edges of one graph are taken as the vertices of a new graph with adjacency preserved, is arguably the most interesting of graph transformations. In this survey, we will begin looking at characterisations of line graphs, focusing first on results related to our set of nine forbidden subgraphs. This will be followed by a discussion of some generalisations of line graphs, including our investigations into the Krausz dimension of a graph G, defined as the minimum, over all partitions of the edge-set of G into complete subgraphs, of the maximum number of subgraphs containing any vertex (the maximum in Krausz's characterisation of line graphs being 2).