Graph decompositions and D₃-paths with a prescribed endvertex

Let k and n be positive integers. Let G be a graph of order n≤4k, and let n = ∑^k_i=1 a_i be a partition of n into k positive integers a_i with 1≤a_i≤4. In Matsunaga (Preprint, 1995) one of the authors proved that if G is 2-connected and the minimum degree of G is at least k, then with certain types of exceptions being allowed, the vertex set of G can be decomposed into k disjoint subsets A_l,...,A_k so that |A_i| = a_i and "the subgraph induced by A_i is connected" for all i, 1≤i≤k. In the present paper we extend this result for connected graphs in general. The paper also features the existence of D3-paths with a prescribed endvertex in a graph. (A D3-path in a graph is a path of the graph whose removal leaves no component of order at least 3.)