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We define relative Ruan invariants that count embedded connected symplecticsubmanifolds which contact a fixed stable symplectic hypersurface V in asymplectic 4-manifold (X,w) at prescribed points with prescribed contact orders(in addition to insertions on X\V) for stable V. We obtain invariants of thedeformation class of (X,V,w). Two large issues must be tackled to define suchinvariants: (1) Curves lying in the hypersurface V and (2) genericity resultsfor almost complex structures constrained to make V pseudo-holomorphic (oralmost complex). Moreover, these invariants are refined to take into accountrim tori decompositions. In the latter part of the paper, we extend thedefinition to disconnected submanifolds and construct relative Gromov-Taubesinvariants.

Relative Ruan and Gromov-Taubes Invariants of Symplectic 4-Manifolds