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In light of recent attempts to extend the Cieliebak-Mohnke approach for constructing Gromov-Witten invariants to positive genera, we compare the absolute and relative Gromov-Witten invariants of compact symplectic manifolds when the symplectic hypersurface contains no relevant holomorphic curves. We show that these invariants are then the same, except in a narrow range of dimensions of the target and genera of the domains, and provide examples when they fail to be the same. * Partiallydoi:10.4310/jsg.2016.v14.n4.a8 fatcat:jcimwbnd3nfwfgedfza2y2twuy