A forced-air heat distribution system in a building must get heat from the central furnace at vertex

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A forced-air heat distribution system in a building must get heat from the central furnace at vertex 1 in the figure to each of the rooms located at the other vertices. It is possible to mount ductwork, assumed unidirectional, along each of the edges shown on the graph. Find a sparsest possible system of ductwork to heat the building.

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