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A phylogenetic network is a graph-theoretical tool that is used by biologiststo represent the evolutionary history of a collection of species. One potentialway of constructing such networks is via a distance-based approach, where oneis asked to find a phylogenetic network that in some way represents a givendistance matrix, which gives information on the evolutionary distances betweenpresent-day taxa. Here, we consider the following question. For which~$k$ areunrooted level-$k$ networks uniquely determined by their distance matrices? Weconsider this question for shortest distances as well as for the case that themultisets of all distances is given. We prove that level-$1$ networks andlevel-$2$ networks are reconstructible from their shortest distances andmultisets of distances, respectively. Furthermore we show that, in general,networks of level higher than~$1$ are not reconstructible from shortestdistances and that networks of level higher than~$2$ are not reconstructiblefrom their multisets of distances.

Reconstructibility of unrooted level-$k$ phylogenetic networks from distances