Geodesic
Remarks on partially square graphs, hamiltonicity and circumference
Discussiones Mathematicae. Graph Theory, Tome 21 (2001) no. 2, pp. 255-266

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Given a graph G, its partially square graph G* is a graph obtained by adding an edge (u,v) for each pair u, v of vertices of G at distance 2 whenever the vertices u and v have a common neighbor x satisfying the condition N_G(x) ⊆ N_G[u] ∪ N_G[v], where N_G[x] = N_G(x) ∪ x. In the case where G is a claw-free graph, G* is equal to G². We define σ°ₜ = min ∑_x∈S d_G(x):S is an independent set in G* and |S| = t. We give for hamiltonicity and circumference new sufficient conditions depending on σ° and we improve some known results.
Keywords: partially square graph, claw-free graph, independent set, hamiltonicity and circumference 
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Kheddouci, Hamamache. Remarks on partially square graphs, hamiltonicity and circumference. Discussiones Mathematicae. Graph Theory, Tome 21 (2001) no. 2, pp. 255-266. http://geodesic.mathdoc.fr/item/DMGT_2001_21_2_a9/