  In this article, we establish a connection between two models for $r$-spin
structures on surfaces: the marked PLCW decompositions of Novak and
Runkel-Szegedy, and the structured graphs of Dyckerhoff-Kapranov. We use these
models to describe $r$-spin structures on open-closed bordisms, leading to a
generators-and-relations characterization of the 2-dimensional open-closed
$r$-spin bordism category. This results in a classification of 2-dimensional
open closed field theories in terms of algebraic structures we term
"knowledgeable $\Lambda_r$-Frobenius algebras". We additionally extend the
state sum construction of closed $r$-spin TFTs from a $\Lambda_r$-Frobenius
algebra $A$ with invertible window element of Novak and Runkel-Szegedy to the
open-closed case. The corresponding knowledgeable $\Lambda_r$-Frobenius algebra
is $A$ together with the $\mathbb{Z}/r$-graded center of $A$.
