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In this paper we formally define the structural description of an object and the concepts of exact and inexact matching of two structural descriptions. We discuss the problems associated with a brute-force backtracking tree search for inexact matching and develop several different algorithms to make the tree search more efficient. We develop the formula for the expected number of nodes in the tree for backtracking alone and with a forward checking algorithm. Finally, we present experimental results showing that forward checking is the most efficient of the algorithms tested.