AI 中文总结
本文证明美丽图等价于无诱导$C_4$、gem、net、watch或奇洞的图，即$C_4$-自由可比图，并给出基于闭邻域构造偏序的证明，带来多项式识别与精确障碍族。
AI 中文摘要
如果一个图的每个诱导子图都是某个二元矩阵的所有极大非空1-矩形的交集图，且邻接关系由公共单元定义，则该图称为美丽的。美丽图作为Berge图的一个遗传类被引入，但此前未获得完整的禁止诱导子图刻画。我们证明：一个有限图是美丽的当且仅当它不含诱导的$C_4$、gem、net、watch或奇洞。更精确地说，这些图正是允许一个偏序使得每个区间都是链的$C_4$-自由可比图。证明通过包含极大闭邻域来构造这样的偏序：其代表元诱导一个二分图，该二分图的支配区域允许相容的定向。随后，一个序矩阵表示该图，并通过其主子矩阵表示每个诱导子图。表示步骤利用经典的双界图以及极大双团与区间交集闭偏序之间的已知对应关系来表述。推论包括多项式时间识别、精确的极小障碍族，以及在$K_4$-自由情形下的修正刻画。
英文摘要
A graph is beautiful if each of its induced subgraphs is the intersection graph of all maximal nonempty 1-rectangles of a binary matrix, with adjacency defined by a common cell. Beautiful graphs were introduced as a hereditary class of Berge graphs, but a complete forbidden-induced-subgraph characterization was not obtained. We prove that a finite graph is beautiful if and only if it has no induced $C_4$, gem, net, watch, or odd hole. More precisely, these graphs are exactly the $C_4$-free comparability graphs admitting a partial order in which every interval is a chain. The proof constructs such an order from inclusion-maximal closed neighbourhoods: their representatives induce a bipartite graph whose domination regions admit compatible orientations. One order matrix then represents the graph and, through its principal submatrices, every induced subgraph. The representation step is formulated using classical double-bound graphs and the established correspondence between maximal bicliques and interval-intersection-closed posets. Consequences include polynomial-time recognition, the exact minimal obstruction families, and a corrected characterization in the $K_4$-free case.
