We consider the shape optimization problem
\hbox{$\min\big\{\E(\Gamma)\ :\ \Gamma\in\A,\ \H^1(\Gamma)=l\
\big\},$}min{ℰ(Γ):Γ ∈ 𝒜,
ℋ1(Γ) = l}, where ℋ1 is the one-dimensional Hausdorff
measure and 𝒜 is an admissible class of one-dimensional sets
connecting some prescribed set of points \hbox{$\D=\{D_1,\dots,D_k\}\subset\R^d$}𝒟 = { D1,...,Dk } ⊂ Rd. The cost functional ℰ(Γ) is the
Dirichlet energy of Γ defined through the Sobolev functions on
Γ vanishing on the points
Di. We analyze the existence of a solution
in both the families of connected sets and of metric graphs. At the end, several explicit
examples are discussed.