Geodesic
Graphs with given neighborhoods of vertices
Matematičeskie zametki, Tome 3 (1968) no. 2, pp. 211-216.

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This article establishes criteria for existence of finite graphs $L$ in which the neighborhoods of all vertices are isomorphic to a given graph $M$ for the cases in which $M$ is a simple arc and a simple cycle. 
@article{MZM_1968_3_2_a11,
     author = {S. Ya. Agakishieva},
     title = {Graphs with given neighborhoods of vertices},
     journal = {Matemati\v{c}eskie zametki},
     pages = {211--216},
     publisher = {mathdoc},
     volume = {3},
     number = {2},
     year = {1968},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1968_3_2_a11/}
}
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S. Ya. Agakishieva. Graphs with given neighborhoods of vertices. Matematičeskie zametki, Tome 3 (1968) no. 2, pp. 211-216. http://geodesic.mathdoc.fr/item/MZM_1968_3_2_a11/