Decomposing a graph into subgraphs with small components
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The component size of a graph is the maximum number of edges in any connected component of the graph. Given a graph G and two integers k and c, (k,c)-Decomposition is the problem of deciding whether G admits an edge partition into k subgraphs with component size at most c. We prove that for any fixed k ≥ 2 and c ≥ 2, (k,c)-Decomposition is NP-complete in bipartite graphs. Also, when both k and c are part of the input, (k,c)-Decomposition is NP-complete even in trees. Moreover, (k,c)-Decomposition in trees is W[1]-hard with parameter k, and is FPT with parameter c. In addition, we present approximation algorithms for decomposing a tree either into the minimum number of subgraphs with component size at most c, or into k subgraphs minimizing the maximum component size. En route to these results, we also obtain a fixed-parameter algorithm for Bin Packing with the bin capacity as parameter.
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