Abstract
Theorems of an ‘if and only if’ nature are very much sought after, and this certainly applies to graph theory. In this chapter, three properties are given that are equivalent to a graph being a line graph. The first involves a partition of the edge set into complete subgraphs. The second involves the fact that the claw K 1,3 is forbidden as an induced subgraph and there are restrictions on the neighborhoods of the vertices of any copy of K 1,1,2 that is an induced subgraph. The third is a list of nine graphs that are forbidden as induced subgraphs. Line graphs of special families of graphs, including trees and other bipartite graphs, are also characterized.
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Beineke, L.W., Bagga, J.S. (2021). Characterizations of Line Graphs. In: Line Graphs and Line Digraphs. Developments in Mathematics, vol 68. Springer, Cham. https://doi.org/10.1007/978-3-030-81386-4_3
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DOI: https://doi.org/10.1007/978-3-030-81386-4_3
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