双色区间图与区间夹层图：容差层级中的两个新图类

中文摘要 AI 辅助

我们引入了两个新的图类，即双色区间图和区间夹层图，它们源于计算机视觉中鲁棒直线拟合问题的研究。在双色区间图中，每个顶点被赋予一个实数区间和两种颜色之一。不同颜色的顶点当且仅当它们的区间相交时相邻，而同颜色的顶点当且仅当一个区间的中心位于另一个区间内时相邻。丢弃着色后得到区间夹层图，它夹在一族区间的中心包含图与相交图之间。我们证明了这两个类的结构结果，确定了它们各自包含哪些洞、反洞、树和完全二分图。因此，这两个类都不能由有限个禁止诱导子图来刻画。我们将这两个类严格置于容差层级中单位容差图与余可比较图之间，将它们与相邻类区分开来。特别地，我们构造了一个无限族的真容差图，它们不是双色区间图，且每个图都具有该性质的最小性。最后，我们考虑这两个类的识别问题。我们证明了这两个类中的每个图都有一个多项式大小的整数表示，因此这两个问题都属于$\ extsf{NP}$。

英文摘要

We introduce two new graph classes, the bicoloured-interval graphs and the interval-sandwich graphs, arising from the study of the robust line fitting problem in computer vision. In a bicoloured-interval graph, each vertex is assigned a real interval and one of two colours. Vertices of different colours are adjacent precisely when their intervals intersect, and vertices of the same colour precisely when the centre of one interval lies in the other. Discarding the colouring yields the interval-sandwich graphs, sandwiched between the $50\%$-tolerance graph and the intersection graph of a family of intervals. We prove structural results for both classes, determining which holes, antiholes, trees, and complete bipartite graphs each contains. Consequently, neither class is characterised by finitely many forbidden induced subgraphs. We place both classes strictly between the unit tolerance graphs and the co-comparability graphs in the tolerance hierarchy, separating them from the neighbouring classes. In particular, we construct an infinite family of proper tolerance graphs, each of which is a minimal forbidden induced subgraph for the bicoloured-interval graphs. Finally, we consider the recognition problems for both classes. We show that every graph in either class has a polynomial-size integer representation, and hence that both problems lie in~$\textsf{NP}$.

发表机构

• Edith Cowan University（伊迪斯考文大学）
• Sun Yat-sen University（中山大学）

机构由 AI 辅助整理，请以论文原文为准。

补充信息

备注

51 pages including references and appendix; 2 figures; streamlined proofs and minor corrections throughout the article;