Graph decompositions and \(D_3\)-paths with a prescribed endvertex (Q1970703)

The paper presents result generalizing a previous work by Matsunaga. The authors show that the vertex set of a connected graph with at most \(4k\) nodes can be decomposed into subsets \(A_1, \ldots,A_k\) such that the subgraph induced by \(A_i\) is connected, provided that \(|A_i|\leq 4\) for all \(i = 1,\ldots, k\), unless the graph belongs to a list of exceptional graphs (Matsunaga had given a similar result for 2-connected graphs). Other types of decompositions are also investigated. The paper also contains a result predicting the existence of a \(D_3\)-path in a connected graph (a path whose removal leaves no component of order at least 3).