  We define combinatorial counterparts to the geometric string vertices of
Sen-Zwiebach and Costello-Zwiebach, which are certain closed subsets of the
moduli spaces of curves. Our combinatorial vertices contain the same
information as the geometric ones, are effectively computable, and act on the
Hochschild chains of a cyclic $A_\infty$-algebra.
  This is the first in a series of two papers where we define enumerative
invariants associated to a pair consisting of a cyclic $A_\infty$-algebra and a
splitting of the Hodge filtration on its cyclic homology. These invariants
conjecturally generalize the Gromov-Witten and Fan-Jarvis-Ruan-Witten
invariants from symplectic geometry, and the Bershadsky-Cecotti-Ooguri-Vafa
invariants from holomorphic geometry.
