Structure-preserving hierarchical decompositions

A hierarchy tree of a graph G is a rooted tree whose leaves are the vertices of G; the internal nodes are usually called clusters. Hierarchy trees are well suited for representing hierarchical decompositions of graphs. In this paper we introduce the notion of P-validity of hierarchy trees with respect to a given graph property P. This notion reflects the similarity between any high-level representation of G obtained from the hierarchy tree and the topological structure of G. Maintaining, the properties of a graph at any level of abstraction is especially relevant in graph drawing applications. We present a structural characterization of P-valid hierarchy trees when the clustered graph is a tree and property P is the acyclicity. Besides being interesting in its own right, our structure theorem can be used in the design of a polynomial time algorithm for recognizing P-valid hierarchy trees.