In this paper, we show that calculating the partition function of optimal medians of binary strings with Hamming distance is #P-complete for several weight functions. The case when the weight function is the factorial function has application in bioinformatics. In that case, the partition function counts the most parsimonious evolutionary scenarios on a star tree under several models in bioinfo...

Let e be an edge of a median graph G which is contained in exactly one 4-cycle. Then it is proved that G n e is a median graph. The converse holds as well provided that e is not contained is a subgraph isomorphic to the 3-cube minus a vertex. These results are used to give several characterizations of median grid graphs. (Grid graphs are subgraphs of complete grids, i.e. of the Cartesian produc...