Windmill

Windmill --- ветряная мельница.

A windmill, denoted by [math]\displaystyle{ pS.K_{r} }[/math], is the graph obtained by coalescing a complete graph [math]\displaystyle{ K_{r} }[/math], [math]\displaystyle{ r \geq 2 }[/math], with [math]\displaystyle{ p }[/math] disjoint copies of a graph [math]\displaystyle{ S(v) }[/math], rooted at [math]\displaystyle{ v \in V(S), \; 0 \leq p \leq r }[/math]. The root vertex, [math]\displaystyle{ v }[/math], in the labelled graph [math]\displaystyle{ S }[/math], is identified with one vertex of [math]\displaystyle{ K_{r} }[/math] so that [math]\displaystyle{ p }[/math] distinct vertices of [math]\displaystyle{ K_{r} }[/math] are cut vertices of the windmill.

A complex windmill is the graph obtained by coalescing a complete graph [math]\displaystyle{ K_{r}, \; r \geq 2 }[/math], with disjoint graphs [math]\displaystyle{ S_{1}, S_{2}, \ldots, S_{p} }[/math] at [math]\displaystyle{ p }[/math] distinct vertices of [math]\displaystyle{ K_{r}, \; 0 \leq p \leq r }[/math], so that these vertices are cut vertices of the complex windmill.