Topological field theories on open-closed $r$-spin surfaces

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 $A$.