Minimum size tree-decompositions

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We study in this paper the problem of computing a tree-decomposition of a graph with width at most k and minimum number of bags. More precisely, we focus on the following problem: given a fixed k≥1, what is the complexity of computing a tree-decomposition of width at most k with minimum number of bags in the class of graphs with treewidth at most k? We prove that the problem is NP-complete for any fixed k≥4 and polynomial for k≤2; for k=3, we show that it is polynomial in the class of trees and 2-connected outerplanar graphs.