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Graphs with prescribed neighbourhood graphs. (English) Zbl 0579.05045
Let u be a vertex in a graph G. Then $$N_ G(u)$$ denotes the subgraph of G induced by the vertices adjacent to u in G. The subgraph $$N_ G(u)$$ is called the neighborhood graph of u in G (also called the link of u in G). The author investigates those graphs G for which $$N_ G(u)$$ is isomorphic to the complement of a path (respectively, cycle) for all vertices u of G. In particular, if G is a graph such that $$N_ G(u)\cong \bar P_ n$$ (n$$\geq 4)$$ for every u, then $$G\cong C_{n+4}$$. Also, there exists a graph G with the property that $$N_ G(u)\cong \bar C_ n$$ for every vertex u if and only if $$3\leq n\leq 6$$.
Reviewer: G.Chartrand

##### MSC:
 05C75 Structural characterization of families of graphs
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##### References:
 [1] Theory of Graph and its Applications. Proc. Symp. Smolenice 1983 by M. Fiedler, Prague 1964. [2] SEDLÁČEK J.: Local properties of graphs. (Czech, English summary.) Časop. pěst. mat. 106, 1981, 290-298. · Zbl 0478.05080 · eudml:21468
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