Circular-arc graphs are the intersection graphs of arcs of a circle. The main result of this work describes the structure of all \emph{normalized intersection models} of circular-arc graphs. Normalized models of a circular-arc graph reflect the neighborhood relation between its vertices and can be seen as its canonical representations; in particular, any intersection model can be made normalized by possibly extending some of its arcs. We~devise a data-structure, called \emph{PQM-tree}, that maintains the set of all normalized models of a circular-arc graph. We show that the PQM-tree of a circular-arc graph can be computed in linear time. Finally, basing on PQM-trees, we provide a linear-time algorithm for the canonization and the isomorphism problem for circular-arc graphs. We describe the structure of the normalized models of circular-arc graphs using an approach proposed by Hsu~[\emph{SIAM J. Comput. 24(3), 411--439, (1995)}]. In the aforementioned work, Hsu claimed the construction of decomposition trees representing the set of all normalized intersection models of circular-arc graphs and an $\mathcal{O}(nm)$ time isomorphism algorithm for this class of graphs. However, the counterexample given in~[\emph{Discrete Math. Theor. Comput. Sci., 15(1), 157--182, 2013}] shows that Hsu's isomorphism algorithm is not incorrect. Also, in a companion paper we show that the decomposition trees proposed by Hsu are not constructed correctly; in particular, we showed that there are circular-arc graphs whose all normalized models do not follow the description given by Hsu.

翻译：圆弧图是圆周上弧的相交图。本工作的主要结果描述了圆弧图所有**规范化相交模型**的结构。圆弧图的规范化模型反映了其顶点间的邻域关系，可视为其规范表示；特别地，任何相交模型都可通过适当延伸某些弧转化为规范化模型。我们设计了一种称为**PQM树**的数据结构，用于维护圆弧图的所有规范化模型。我们证明圆弧图的PQM树可在线性时间内计算得到。最后，基于PQM树，我们为圆弧图的规范化表示与同构判定问题提供了线性时间算法。我们采用Hsu[《SIAM计算杂志》24(3), 411--439, (1995)]提出的方法描述圆弧图规范化模型的结构。在该研究中，Hsu声称构建了表示圆弧图所有规范化相交模型的分解树，并为该类图提供了$\mathcal{O}(nm)$时间的同构判定算法。然而，[《离散数学与理论计算机科学》15(1), 157--182, 2013]中给出的反例表明Hsu的同构算法存在错误。此外，在一篇配套论文中我们证明Hsu提出的分解树构建并不正确；特别地，我们指出存在某些圆弧图，其所有规范化模型均不符合Hsu给出的结构描述。