  The adjunction inequality is a key tool for bounding the genus of smoothly
embedded surfaces in 4-manifolds. Using gauge-theoretic invariants, many
versions of this inequality have been established for both closed surfaces and
surfaces with boundary. However, these invariants generally require some global
geometry, such as a symplectic structure or nonzero Seiberg-Witten invariants.
In this paper, we extend previous work on trisections and the Thom conjecture
to obtain adjunction information in a much larger class of smooth 4-manifolds.
We intrdouce polyhedral decompositions of almost-complex 4-manifolds and give a
criterion in terms of this decomposition for surfaces to satisfy the adjunction
inequality.
