Individualization-Refinement (IR) algorithms form the standard method and currently the only practical method for symmetry computations of graphs and combinatorial objects in general. Through backtracking, on each graph an IR-algorithm implicitly creates an IR-tree whose order is the determining factor of the running time of the algorithm. We give a precise and constructive characterization which trees are IR-trees. This characterization is applicable both when the tree is regarded as an uncolored object but also when regarded as a colored object where vertex colors stem from a node invariant. We also provide a construction that given a tree produces a corresponding graph whenever possible. This provides a constructive proof that our necessary conditions are also sufficient for the characterization.

中文翻译：

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个性化-细化树的表征

个性化-细化 (IR) 算法形成了标准方法，并且是目前用于图形和组合对象对称性计算的唯一实用方法。通过回溯，在每个图上，IR 算法隐式地创建了一个 IR 树，其顺序是算法运行时间的决定因素。我们给出了哪些树是 IR 树的精确和建设性的特征。当树被视为无色对象时，以及当将树视为顶点颜色源自节点不变性的有色对象时，这种表征都适用。我们还提供了一种结构，只要可能，给定一棵树就会生成相应的图。这提供了一个建设性的证据，证明我们的必要条件对于表征也是充分的。