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arXiv · 2609.12293

Bicoloured-interval and interval-sandwich graphs: two new classes in the tolerance hierarchy

Abdul Basit·David Suter·Erchuan Zhang
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Abstract

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~$\textsf{NP}$.

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BibTeXRIS

Abdul Basit, David Suter, Erchuan Zhang. 2026-10-01. Bicoloured-interval and interval-sandwich graphs: two new classes in the tolerance hierarchy. https://arxiv.org/abs/2609.12293

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