A unified FPT Algorithm for Width of Partition Functions

During the last decades, several polynomial-time algorithms have been designed that decide if a graph has treewidth (resp., pathwidth, branchwidth, etc.) at most $k$, where $k$ is a fixed parameter. Amini {\it et al.} (to appear in SIAM J. Discrete Maths.) use the notions of partitioning-trees and partition functions as a generalized view of classical decompositions of graphs, namely tree-decomposition, path-decomposition, branch-decomposition, etc. In this paper, we propose a set of simple sufficient conditions on a partition function $\Phi$, that ensures the existence of a linear-time explicit algorithm deciding if a set $A$ has $\Phi$-width at most $k$ ($k$ fixed). In particular, the algorithm we propose unifies the existing algorithms for treewidth, pathwidth, linearwidth, branchwidth, carvingwidth and cutwidth. It also provides the first Fixed Parameter Tractable linear-time algorithm deciding if the $q$-branched treewidth, defined by Fomin {\it et al.} (Algorithmica 2007), of a graph is at most $k$ ($k$ and $q$ are fixed). Our decision algorithm can be turned into a constructive one by following the ideas of Bodlaender and Kloks (J. of Alg. 1996).