Let n-1, n2, ⋯, nk be integers, n = σ ni, ni ≥ 3, and let for each 1 ≤ i ≤ k, Hi be a cycle or a tree on ni vertices. We prove that every graph G of order at least n with σ2(G) ≥2(n - k) - 1 contains k vertex disjoint subgraphs H′1, H′2, ⋯, H′k, where H′ = Hi, if H i is a tree, and H′i is a cycle with ni - 3 chords incident with a common vertex, if Hi is a cycle. © 2008 Wiley Periodicals, Inc.