  Gallai's path decomposition conjecture states that if $G$ is a connected
graph on $n$ vertices, then the edges of $G$ can be decomposed into at most
$\lceil \frac{n }{2} \rceil$ paths. A graph is said to be an odd semi-clique if
it can be obtained from a clique on $2k+1$ vertices by deleting at most $k-1$
edges. Bonamy and Perrett asked if the edges of every connected graph $G$ on
$n$ vertices can be decomposed into at most $\lfloor \frac{n}{2} \rfloor$ paths
unless $G$ is an odd semi-clique. A graph $G$ is said to be 2-degenerate if
every subgraph of $G$ has a vertex of degree at most $2$. In this paper, we
prove that the edges of any connected 2-degenerate graph $G$ on $n$ vertices
can be decomposed into at most $\lfloor \frac{n }{2} \rfloor$ paths unless $G$
is a triangle.
