Graph decomposition of slim graphs

A Graph G = (V, E) is called k-slim if for every subgraph S = (V_S, E_S) of G with s = |V_S| ≥ k there exists K ⊂ V_S, |K| = k, such that the vertices of V_S\K can be partitioned into two subsets, A and B, such that |A| ≤ 2/3s and |B| ≤ 2/3s and no edge of E_S connects a vertex from A and a vertex from B. k-slim graphs contain, in particular, the graphs with tree-width k. In this paper we give an algorithm solving the H-decomposition problem for a large family of graphs H which contains, among others, the stars, the complete graphs, and the complete r-partite graphs where r ≥ 3. The algorithm runs in polynomial time in case the input graph is k-slim, where k is fixed. In particular, our algorithm runs in polynomial time when the input graph has bounded tree width k. Our results supply the first polynomial time algorithm for H-decomposition of connected graphs H having at least 3 edges, in graphs with bounded tree-width.