A path decomposition of a graph G is a list of paths such that each edge appears in exactly one path in the list. G is said to admit a {Pl }-decomposition if G can be decomposed into some copies of Pl, where Pl is a path of length l - 1. Similarly, G is said to admit a {Pl, Pk}-decomposition if G can be decomposed into some copies of Pl or Pk. An k-cycle, denoted by Ck, is a cycle with k vertices. An odd tree is a tree of which all vertices have odd degree. In this paper, it is shown that a connected graph G admits a {P3, P4}-decomposition if and only if G is neither a 3-cycle nor an odd tree. This result includes the related result of Yan, Xu and Mutu. Moreover, two polynomial algorithms are given to find {P3}-decomposition and {P3, P4}-decomposition of graphs, respectively. Hence, {P3}-decomposition problem and {P3, P4}-decomposition problem of graphs are solved completely.
