  A sequence of constant mean curvature surfaces $\Sigma_j$ with mean curvature
$H_j \to \infty$ in a three-dimensional manifold $M$ condenses to a compact and
connected graph $\Gamma$ consisting of a finite union of curves if $\Sigma_j$
is contained in a tubular neighbourhood of $\Gamma$ of size $\mathcal O(1/H_j)$
for every $j \in \N$. This paper gives sufficient conditions on $\Gamma$ for
the existence of a sequence of compact, embedded constant mean curvature
surfaces condensing to $\Gamma$. The conditions are: each curve in $\gamma$ is
a critical point of a functional involving the scalar curvature of $M$ along
$\gamma$; and each curve must satisfy certain regularity, non-degeneracy and
boundary conditions. When these conditions are satisfied, the surfaces
$\Sigma_j$ can be constructed by gluing together small spheres of radius
$2/H_j$ positioned end-to-end along the edges of $\Gamma$.
