  Kontsevich introduced certain ribbon graphs as cell decompositions for
combinatorial models of moduli spaces of complex curves with boundaries in his
proof of Witten's conjecture. In this work, we define four types of generalised
Kontsevich graphs and find combinatorial relations among them. We call the main
type ciliated maps and use the auxiliary ones to show they satisfy a Tutte
recursion that we turn into a combinatorial interpretation of the loop
equations of topological recursion for a large class of spectral curves. It
follows that ciliated maps, which are Feynman graphs for the Generalised
Kontsevich matrix Model (GKM), are computed by topological recursion. Our
particular instance of the GKM relates to the r-KdV integrable hierarchy and
since the string solution of the latter encodes intersection numbers with
Witten's $r$-spin class, we find an identity between ciliated maps and $r$-spin
intersection numbers, implying that they are also governed by topological
recursion. In turn, this paves the way towards a combinatorial understanding of
Witten's class. This new topological recursion perspective on the GKM provides
concrete tools to explore the conjectural symplectic invariance property of
topological recursion for large classes of spectral curves.
