Edge search in graphs with restricted test sets

Abstract

Suppose a graph G ( V , E ) contains one defective edge e . We search for the endpoints of e by asking questions of the form “Is at least one of the vertices of X an endpoint of e ?”, where X is a subset of V with cardinality at most p . Then what is the minimum number c p ( G ) of questions, which are needed in the worst case to find e ? We solve this search problem suggested by M. Aigner in [M. Aigner, Combinatorial Search, Teubner, 1988] by deriving lower and sharp upper bounds for c p ( G ) . For the case that G is the complete graph K n the problem described above is equivalent to the ( 2 , n ) group testing problem with test sets of cardinality at most p . We present sharp upper and lower bounds for the worst case number c p of tests for this group testing problem and show that the maximum difference between the upper and the lower bounds is 3.