We consider graphs, which and all induced subgraphs of which possess the following property: the maximum number of disjoint paths on k vertices equals the minimum cardinality of vertex sets, covering all paths on k vertices. We call such graphs König for the k-path and all its spanning supergraphs. For each odd k, we reveal an infinite family of minimal forbidden subgraphs for them. Additionally, for every odd k, we present a procedure for constructing some of such graphs, based on the operations of adding terminal subgraphs and replacement of edges with subgraphs.

中文翻译：

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关于奇数路径及其所有跨越超图的 König 图的部分描述

我们考虑图，其及其所有诱导子图具有以下性质：k个顶点上不相交路径的最大数量等于顶点集的最小基数，涵盖k个顶点上的所有路径。我们将k路径及其所有跨越超图的此类图称为 König 。对于每个奇数k，我们为它们揭示了一个无限的最小禁止子图族。此外，对于每个奇数k，我们基于添加终端子图和用子图替换边的操作，提出了构建一些这样的图的过程。