What Is between Chordal and Weakly Chordal Graphs?

An (h, s, t)-representation of a graph G consists of a collection of subtrees {Sv |v ∈ V (G)} of a tree T , such that (i) the maximum degree of T is at most h, (ii) every subtree has maximum degree at most s, and (iii) there is an edge between two vertices in the graph if and only if the corresponding subtrees in T have at least t vertices in common. For example, chordal graphs correspond to [∞,∞, 1] = [3, 3, 1] = [3, 3, 2] graphs (notation of ∞ here means that no restriction is imposed). We investigate the complete bipartite graph K2,n and prove new theorems characterizing thoseK2,n graphs that have an (h, s, 2)-representation and those that have an (h, s, 3)-representation. We characterize [3, 2, 4] graphs as equivalent to the 4-flower-free [2, 4, 4] graphs and give a recognition algorithm for [2, 3, 4] graphs. Based on these characterizations, we present new results that confirm that weakly chordal graphs, as opposed to chordal graphs, can not be characterized within the [h, s, t] framework. Furthermore, we show a hierarchy of families of graphs between chordal and weakly chordal within the [h, s, t] framework.