Caterpillar Tolerance Representations

Various families of tolerance graphs of subtrees for specific families of host trees and tolerance functions have been successfully characterized. For example, chordal graphs are intersection (tolerance 1) graphs of subtrees of general trees, see [2], [5], and [8]. Intersection graphs of subtrees of a path are those that are chordal and do not contain an asteroidal triple, see [7]. We denote by cat[h, t] the tolerance graphs of subtrees where the host is a caterpillar of maximum degree h and the tolerance function is the constant t for all vertices. We give a characterization involving asteroidal triples with no simplicial vertices for the equivalent families cat[3,1], cat[3,2], and cat[h,1] for all h ≥ 3. We also prove that cat[4,2] = cat[3,3] and that cat[h,2] ( cat[h− 1,3] for h ≥ 5.