Partially square graph

Partially square graph --- частично квадратный граф.

Given a graph [math]\displaystyle{ G }[/math], its partially square graph [math]\displaystyle{ G^{\ast} }[/math] is the graph obtained by adding an edge [math]\displaystyle{ (u,v) }[/math] for each pair of vertices of [math]\displaystyle{ G }[/math] at distance 2, whenever the vertices [math]\displaystyle{ u }[/math] and [math]\displaystyle{ v }[/math] have a common neighbor [math]\displaystyle{ x }[/math] satisfying the condition

[math]\displaystyle{ N_{G}(x) \subseteq N_{G}[u] \cup N_{G}[v] }[/math]

where N(x) is an open neighborhood and N[x] is a closed neighborhood of a vertex [math]\displaystyle{ x }[/math]. In the case where [math]\displaystyle{ G }[/math] is a claw-free graph, [math]\displaystyle{ G^{\ast} }[/math] is equal to [math]\displaystyle{ G^{2} }[/math].