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Neighbourhoods in line graphs. (English) Zbl 0766.05071
L. W. Beineke’s characterization [J. Comb. Theory 9, 129-135 (1970; Zbl 0202.557)] of line graphs by 9 forbidden induced subgraphs is now very well known. Since four of these obstructions have a vertex adjacent to all other vertices, forbidding these subgraphs means just a restriction on all the neighborhoods of the graph. Such graphs obeying these neighborhood conditions are called locally-\(\Theta\) graphs; they can be expressed in a very neat form, as is shown in the paper. Furthermore, for a graph \(H\), a locally-\(H\) graph is one where all neighborhoods are isomorphic to \(H\). It is shown that the question whether a locally-\(H\) graph is a line graph or not only depends on \(H\).

05C75 Structural characterization of families of graphs
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