Subtree and substar intersection numbers (Q686262)

The authors define the tree number of a graph \(G\) to be the minimum \(t\) such that \(G\) has an intersection representation in which to each vertex of \(G\) a union of \(t\) subtrees of an underlying tree is assigned. In the case all subtrees are stars they speak of the star number. A variety of results on the introduced invariants are presented, mostly bounds in terms of the clique, the interval numbers, etc.