Essential weakly sparse families for treewidth

Determine whether there exists a weakly sparse family of graphs whose hereditary closure is a minimal hereditary class of unbounded treewidth, equivalently, characterize whether such a family can be essential for treewidth beyond families consisting solely of complete graphs or solely of complete bipartite graphs.

Background

The paper studies essentiality not only for individual graphs but also for families of graphs. It explains that a family is essential for treewidth exactly when its hereditary closure is a minimal hereditary class of unbounded treewidth.

The authors show that families containing infinitely many complete graphs, or infinitely many complete bipartite graphs, are essential, whereas adding a non-clique or non-biclique respectively destroys essentiality. They therefore reduce the unresolved cases to weakly sparse families and identify the existence of an essential weakly sparse family as the central remaining question.