Figure 2

Priorities of all pathgroups of form [(x, a), (x, b), (x, c)] for inserting red edges, for each ancestral vertex in the median problem. Includes sketch of three paths in “x” pathgroup plus other paths involved in calculating priority. For example, completing the pathgroup [(x, y), (x, y), (x, z)] by adding the red edge xy always produces two cycles, but can set up a pathgroup with 3 potential cycles (priority 2), 2 potential cycles (priority 3) or 1 potential cycles (priority 4).