### Article

## On partial descriptions of König graphs for odd paths and all their spanning supergraphs

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.