Leaf Powers and Their Properties: Using the Trees

Abstract

A graph G on n vertices is a k-leaf power ($G \in {\cal G}_{k}$) if it is isomorphic to a graph that can be “generated” from a tree T that has n leaves, by taking the leaves to represent vertices of G, and making two vertices adjacent if and only if they are at distance at most k in T. We address two questions in this paper:

(1) As k increases, do we always have ${\cal G}_{k} \subseteq {\cal G}_{k+1}$ ? Answering an open question of Andreas Brandstädt and Van Bang Le [2,3,1], we show that the answer, perhaps surprisingly, is “no.”

(2) How should one design algorithms to determine, for k-leaf powers, if they have some property?

One way this can be done is to use the fact that k-leaf powers have bounded cliquewidth. This fact, plus the FPT cliquewidth approximation algorithm of Oum and Seymour [14], combined with the results of Courcelle, Makowsky and Rotics [7], allows us to conclude that properties expressible in a general logic formalism, can be decided in FPT time for k-leaf powers, parameterizing by k. This is wildly inefficient. We explore a different approach, under the assumption that a generating tree is given with the graph. We show that one can use the tree directly to decide the property, by means of a finite-state tree automaton. (A more general theorem has been independently obtained by Blumensath and Courcelle [5].)

We place our results in a general context of “tree-definable” graph classes, of which k-leaf powers are one particular example.

This work is supported by the Research Council of Norway.