Graph Decomposition of Slim Graphs

Abstract.

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|≤⅔s and |B|≤⅔s 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.