Inferring a Graph from Path Frequency

We consider the problem of inferring a graph (and a sequence) from the numbers of occurrences of vertex-labeled paths, which is closely related to the pre-image problem for graphs in machine learning: to reconstruct a graph from its feature space representation. We show that this problem can be solved in polynomial time in the size of an output graph if graphs are trees of bounded degree and the lengths of given paths are bounded by a constant. On the other hand, we show that this problem is strongly NP-hard even for planar graphs of bounded degree.

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