Remarks on partially square graphs, hamiltonicity and circumference

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 ${\displaystyle N_G\left(x\right)\subseteq N_G\left[u\right]\cup N_G\left[v\right]}$, where ${\displaystyle N_G\left[x\right]=N_G\left(x\right)\cup \left\{x\right\}}$. In the case where G is a claw-free graph, G* is equal to G². We define ${\displaystyle \sigma °ₜ=min\{{\sum }_{x\in S}{d}_G\left(x\right):Sisan\in dependentset\in G\cdot \text{and}\left|S\right|=t\}}$. We give for hamiltonicity and circumference new sufficient conditions depending on σ° and we improve some known results.