SIAM Journal on Discrete Mathematics, Volume 35, Issue 2, Page 707-751, January 2021.
In 1969, Tucker characterized proper circular-arc graphs as those graphs whose augmented adjacency matrices have the circularly compatible ones property. Moreover, he also found a polynomial-time algorithm for deciding whether any given augmented adjacency matrix has the circularly compatible ones property. These results led to the first polynomial-time recognition algorithm for proper circular-arc graphs. However, as remarked there, this work did not solve the problems of finding a structure theorem and an efficient recognition algorithm for the circularly compatible ones property in arbitrary matrices (i.e., not restricted to augmented adjacency matrices only). In the present work, we solve these problems. More precisely, we give a minimal forbidden submatrix characterization for the circularly compatible ones property in arbitrary matrices and a linear-time recognition algorithm for the same property. We derive these results from analogous ones for the related $D$-circular property. Interestingly, these results lead to a minimal forbidden induced subgraph characterization and a linear-time recognition algorithm for proper circular-arc bigraphs, solving a problem first posed by Basu et al. [J. Graph Theory, 73 (2013), pp. 361--376]. Our findings generalize some known results about $D$-interval hypergraphs and proper interval bigraphs.

中文翻译：

---

圆兼容的，$D$-Circularity 和适当的圆弧双图

SIAM 离散数学杂志，第 35 卷，第 2 期，第 707-751 页，2021 年 1 月。
1969 年，Tucker 将真圆弧图描述为那些增广邻接矩阵具有循环兼容特性的图。此外，他还发现了一种多项式时间算法，用于确定任何给定的增广邻接矩阵是否具有循环兼容的属性。这些结果导致了第一个用于正确圆弧图的多项式时间识别算法。然而，正如那里所说，这项工作并没有解决在任意矩阵中找到结构定理和有效识别算法的问题（即，不仅限于增广邻接矩阵）。在目前的工作中，我们解决了这些问题。更确切地说，我们给出了任意矩阵中循环兼容的属性的最小禁止子矩阵表征和相同属性的线性时间识别算法。我们从相关的 $D$-circular 属性的类似结果中得出这些结果。有趣的是，这些结果导致了最小禁止诱导子图表征和适当圆弧双图的线性时间识别算法，解决了 Basu 等人首先提出的问题。[J. 图论，73 (2013)，第 361--376 页]。我们的发现概括了一些关于 $D$-区间超图和适当区间双图的已知结果。这些结果导致了最小禁止诱导子图表征和用于适当圆弧双图的线性时间识别算法，解决了 Basu 等人首先提出的问题。[J. 图论，73 (2013)，第 361--376 页]。我们的发现概括了一些关于 $D$-区间超图和适当区间双图的已知结果。这些结果导致了最小禁止诱导子图表征和用于适当圆弧双图的线性时间识别算法，解决了 Basu 等人首先提出的问题。[J. 图论，73 (2013)，第 361--376 页]。我们的发现概括了一些关于 $D$-区间超图和适当区间双图的已知结果。