The complete inclusion structure of leaf power classes
Theoretical Computer Science
Characterising (k,l)-leaf powers
Discrete Applied Mathematics
On relaxing the constraints in pairwise compatibility graphs
WALCOM'12 Proceedings of the 6th international conference on Algorithms and computation
Exploring pairwise compatibility graphs
Theoretical Computer Science
Hi-index | 0.00 |
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.