On the König Graphs for a $$\boldsymbol 5 $$-Path and Its Spanning Supergraphs

Abstract

We describe the hereditary class of graphs whose every subgraph has the property that the maximum number of disjoint $$5$$ -paths (paths on $$5 $$ vertices) is equal to the minimum size of the sets of vertices having nonempty intersection with the vertex set of each $$5 $$ -path. We describe this class in terms of the “forbidden subgraphs” and give an alternative description, using some operations on pseudographs.