The graph decomposition problem consists of dividing a graph into components, patterns or partitions which satisfy some specifications. In this paper,we give interest to graph decomposition into particular patterns: disjoint paths of length two. We present the first Self-stabilizing algorithm for finding a Maximal Decomposition of an arbitrary graph into disjoint Paths of length two (SMDP). Then, we give the correctness proof and we show that SMDP converges in $O(\Delta m)$ moves where $m$ is the number of edges and $\Delta$ the maximum degree in the graph~$G$.