Abstract
Consider the following generalization of the sequential group testing problem for 2 defective items, which is suggested by Aigner (1988) in [1]: Suppose a graph G contains one defective edge e ∗ . Find the endpoints of e ∗ by testing whether a subset of vertices of cardinality at most 2 contains at least one of the endpoints of e ∗ or not. What is then the minimum number c 2 ( G ) of tests, which are needed in the worst case to identify e ∗ ? In Gerzen (2009) [10], this problem was partially solved by deriving sharp lower and upper bounds for c 2 ( G ) . In addition, it was proved that the determination of c 2 ( G ) is an NP-complete problem. Among others, it was shown that the inequality | E | ≤ 4 ( c 2 − 1 2 ) + 4 = 2 c 2 2 − 6 c 2 + 8 holds for graphs with 2-complexity c 2 and the edge set E . In the present paper, we study the class of graphs for which this inequality is sharp and characterize these graphs in several ways. We suppose that for those graphs the 2-complexity can be computed in polynomial time by means of this characterization.
Published Version
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