Detecting tampering in a random hypercube

Consider the random hypercube H 2 (pn) obtained from the hypercube H n 2 by deleting any given edge with probabilty 1− pn, independently of all the other edges. A diameter path in H 2 is a longest geodesic path in H n 2 . Consider the following two ways of tampering with the random graph H 2 (pn): (i) choose a diameter path at random and adjoin all of its edges to H 2 (pn); (ii) choose a diameter path at random from among those that start at 0 = (0, · · · , 0), and adjoin all of its edges to H 2 (pn). We study the question of whether these tamperings are detectable asymptotically as n→∞.