A ternary search problem on graphs

Abstract

In a previous paper, Aigner studied the following search problem on graphs. For a graph G, let e ∗ϵE(G) be an unknown edge. In order to find e ∗ , we choose a sequence of test-sets A⊆ V( G) where after every test we are told whether e ∗ has both end-vertices in A, one end-vertex, or none. Find the minimum c( G) of tests required. Since in this problem ternary tests are performed, we have the usual information theoretic bound ⌜log 3|E(G)⌝≤c(G) . Beside his main results which are on complete and complete bipartite graphs, Aigner proved that each forest F with maximum degree at most two is optimal, i.e., the information theoretic bound is achieved. In the present paper we consider the more general question, how close we can come to achieving the information theoretic bound for forests with maximum degree at most r, r=1,2,…. Let F r be the class of forests with non-empty edge-set and maximum degree at most r. We shall investigate the function f(r)=max{c(F)-⌜log 3|E(F)|⌝:Fϵ F r} and obtain the result that f(r)=t+1-⌜log 3(2 t+1)⌝ for 2 t < r≤2 t+1 , t=0,1,…. In addition, we show that, with the exception of five small graphs, all members of F 3 are optimal, and we conjecture that a similar result holds for F r, r≥4 .