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We extend Nekrashevych's $KK$-duality for $C^*$-algebras of regular,recurrent, contracting self-similar group actions to regular, contractingself-similar groupoid actions on a graph, removing the recurrence conditionentirely and generalising from a finite alphabet to a finite graph.  More precisely, given a regular and contracting self-similar groupoid $(G,E)$acting faithfully on a finite directed graph $E$, we associate two$C^*$-algebras, $\mathcal{O}(G,E)$ and $\widehat{\mathcal{O}}(G,E)$, to it andprove that they are strongly Morita equivalent to the stable and unstableRuelle C*-algebras of a Smale space arising from a Wieler solenoid of theself-similar limit space. That these algebras are Spanier-Whitehead dual in$KK$-theory follows from the general result for Ruelle algebras of irreducibleSmale spaces proved by Kaminker, Putnam, and the last author.

KK-duality for self-similar groupoid actions on graphs