No! An Euler circuit would not provide a good solution since the primary goal is simply visiting each vertex rather than traveling each edge. In this problem it would be very inefficient to require each edge to be traveled since this would force multiple visits to the same vertex.

In general, Euler circuits/paths are not the appropriate tool for analyzing problems where it is only important to visit each vertex. For problems of this type, whether an edge is traveled is not important.(Remember, Euler circuits/paths deal with situations where it is important to travel every edge.)

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