Abstract

We say that a first order sentence A defines a graph G if A is true on G but false on any graph non-isomorphic to G. Let L ) denote the minimum length of such a sentence. We define the succinctness function s ) to be the minimum L ) over all graphs on n vertices.We prove that s and q may be so small that for no general recursive function f we can have f)≥n for all n. However, for the function q*=maxi≤nq, which is the least nondecreasing function bounding q from above, we have q*=)log*n, where log*n equals the minimum number of iterations of the binary logarithm sufficient to lower n to 1 or below.We show an upper bound q

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