  Consider a finite connected $2$-complex $X$ endowed with a piecewise
Riemannian metric and whose fundamental group is freely indecomposable, of rank
at least $3$, and in which every $2$-generated subgroup is free. In this paper
we show that we can always find a connected graph $\Gamma\subset X$ such that
$\pi_1 \Gamma\simeq {\mathbb F}_2 \hookrightarrow\pi_1 X$ (in short, a
$2$-incompressible graph) whose length satisfies the following curvature-free
inequality: $\ell(\Gamma)\leq 4\sqrt{2\text{Area}(X)}$. This generalizes a
previous inequality proved by Gromov for closed Riemannian surfaces with
negative Euler characteristic. As a consequence we obtain that the volume
entropy of such $2$-complexes with unit area is always bounded away from zero.
