A graph H is a square root of a graph G, or equivalently, G is the square of H, if G can be obtained from H by adding an edge between any two vertices in H that are of distance 2. The Square Root problem is that of deciding whether a given graph admits a square root. The problem of testing whether a graph admits a square root which belongs to some specified graph class \(\mathcal {H}\) is called the \(\mathcal {H}\)-Square Root problem. By showing boundedness of treewidth we prove that Square Root is polynomial-time solvable on some classes of graphs with small clique number and that \(\mathcal {H}\)-Square Root is polynomial-time solvable when \(\mathcal {H}\) is the class of cactuses.

中文翻译：

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在多项式时间内求仙人掌根。

图H是图G的平方根，或者等效地，如果G可以通过在H中距离为 2的任意两个顶点之间添加边来从H获得，则G是H 的平方。平方根问题是确定给定图是否允许平方根。测试图是否存在属于某个指定图类\(\mathcal {H}\) 的平方根的问题称为\(\mathcal {H}\) -平方根问题。通过显示树宽的有界性，我们证明平方根在一些具有小团数的图类上是多项式时间可解的，并且\(\mathcal {H}\) -当\(\mathcal {H}时，平方根是多项式时间可解的}\)是仙人掌类。