The graph state formalism offers strong connections between quantum
information processing and graph theory. Exploring these connections, first we
show that any graph is a pivot-minor of a planar graph, and even a pivot minor
of a triangular grid. Then, we prove that the application of measurements in
the (X,Z)-plane over graph states represented by triangular grids is a
universal measurement-based model of quantum computation. These two results are
in fact two sides of the same coin, the proof of which is a combination of
graph theoretical and quantum information techniques
