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 Φ, that ensures the existence of a linear-time explicit algorithm deciding if a set A has Φ-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)
