LAUSR

Let $${\mathcal {G}} = \{G_1 = (V, E_1), \ldots , G_m = (V, E_m)\}$$ G = { G 1 = ( V , E 1 ) , … , G m = ( V , E m ) } be a collection of m graphs defined on a common set of vertices V but with different edge sets $$E_1, \ldots , E_m$$ E 1 , … , E m . Informally, a function $$f :V \rightarrow {\mathbb {R}}$$ f : V → R is smooth with respect to $$G_k = (V,E_k)$$ G k = ( V , E k ) if $$f(u) \sim f(v)$$ f ( u ) ∼ f ( v ) whenever $$(u, v) \in E_k$$ ( u , v ) ∈ E k . We study the problem of understanding whether there exists a nonconstant function that is smooth with respect to all graphs in $${\mathcal {G}}$$ G , simultaneously, and how to find it if it exists.