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In this paper we estimate from above the area of the graph of a singular map$u$ taking a disk to three vectors, the vertices of a triangle, and jumpingalong three $\mathcal{C}^2-$ embedded curves that meet transversely at only onepoint of the disk. We show that the relaxed area can be estimated from above bythe solution of a Plateau-type problem involving three entangled nonparametricarea-minimizing surfaces. The idea is to "fill the hole" in the graph of thesingular map with a sequence of approximating smooth two-codimensional surfacesof graph-type, by imagining three minimal surfaces, placed vertically over thejump of $u$, coupled together via a triple point in the target triangle. Such aconstruction depends on the choice of a target triple point, and on aconnection passing through it, which dictate the boundary condition for thethree minimal surfaces. We show that the singular part of the relaxed area of$u$ cannot be larger than what we obtain by minimizing over all possible targettriple points and all corresponding connections.

On the relaxed area of the graph of discontinuous maps from the plane to the plane taking three values with no symmetry assumptions