Minimum Size Tree-Decompositions

Abstract

International audienceWe 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