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=∑i=1kai be a partition of n into k positive integers ai with 1⩽ai⩽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 A1,…,Ak so that |Ai|=ai and “the subgraph induced by Ai 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.)