  Let $\mathcal{G} = \{G_1 = (V, E_1), \dots, G_m = (V, E_m)\}$ be a collection
of $m$ graphs defined on a common set of vertices $V$ but with different edge
sets $E_1, \dots, E_m$. Informally, a function $f :V \rightarrow \mathbb{R}$ is
smooth with respect to $G_k = (V,E_k)$ if $f(u) \sim f(v)$ whenever $(u, v) \in
E_k$. We study the problem of understanding whether there exists a nonconstant
function that is smooth with respect to all graphs in $\mathcal{G}$,
simultaneously, and how to find it if it exists.
