Searching symmetric networks with Utilitarian-Postman paths

We introduce the notion of a Utilitarian Postman (UP) path on a network Q as one which minimizes the expected time required to find a random (uniformly distributed) point, and show that UP paths must be used in a minimax search of a symmetric network. For any network Q, one may consider the zero-sum search game T(Q) in which the (minimizing) Searcher picks a unit speed path S(t) in Q, the Hider picks a point H in Q, and the payoff is the meeting time T = min {t : S(t) = H}. We show first that if Q is symmetric (edge and vertex transitive), then it is optimal for the Hider to pick H uniformly in Q, so that the Searcher must follow a UP path. We then show that if Q is symmetric of odd degree, with n vertices and m unit length edges, the value V of T(Q) satisfies V ≥m/2 + n²-2n/8m, with equality if and only if (^*): Q has a path P = v₁, v₂,..., v_n-1 of distinct vertices, such thatthe edge set Q=Q-U^(n-2)/2_i=1(v_2i, v_2i+1) is connected. In this case, there is a UP path for Q consisting of P followed by an Eulerian path E of Q'. The condition ( ^*) is satisfied by many symmetric graphs, including all complete graphs, complete bipartite graphs, hypercube graphs, high valency graphs, and the Petersen graph. We know of no odd degree symmetric graph not satisfying (^*).