  A long-standing conjecture asserts that there exists a constant $c>0$ such
that every graph of order $n$ without isolated vertices contains an induced
subgraph of order at least $cn$ with all degrees odd. Scott (1992) proved that
every graph $G$ has an induced subgraph of order at least $|V(G)|/(2\chi(G))$
with all degrees odd, where $\chi(G)$ is the chromatic number of $G$, this
implies the conjecture for graphs with { bounded} chromatic number. But the
factor $1/(2\chi(G))$ seems to be not best possible, for example, Radcliffe and
Scott (1995) proved $c=\frac 23$ for trees, Berman, Wang and Wargo (1997)
showed that $c=\frac 25$ for graphs with maximum degree $3$, so it is
interesting to determine the exact value of $c$ for special family of graphs.
In this paper, we further confirm the conjecture for graphs with treewidth at
most 2 with $c=\frac{2}{5}$, and the bound is best possible.
