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Multi-layer graphs can capture qualitatively different types of connectionbetween entities, and networks of this kind are prevalent in biological andsocial systems: for example, a social contact network typically involves bothvirtual and face-to-face interactions between individuals. Since each layer islikely to exhibit stronger and/or more easily identifiable structurallyproperties than the overall system, it is natural to ask whether we can exploitthe structural properties of individual layers to solve NP-hard problemsefficiently on the overall network. In this paper we provide a completecharacterisation of the structural properties required in each layer toguarantee the existence of an FPT algorithm to solve problems definable ineither first-order or monadic second-order logic on the overall system, subjectto the assumption that the structural properties are preserved under deletionof vertices and/or edges.

Two dichotomies for model-checking in multi-layer structures