The Complexity of Conjunctive Queries with Degree 2

It is well known that the tractability of conjunctive query answering can be characterised in terms of treewidth when the problem is restricted to queries of bounded arity. We show that a similar characterisation also exists for classes of queries with unbounded arity and degree 2. To do so we introduce hypergraph dilutions as an alternative method to primal graph minors for studying substructures of hypergraphs. Using dilutions we observe an analogue to the Excluded Grid Theorem for degree 2 hypergraphs. In consequence, we show that that the tractability of conjunctive query answering can be characterised in terms of generalised hypertree width. A similar characterisation is also shown for the corresponding counting problem. We also generalise our main structural result to arbitrary bounded degree and discuss possible paths towards a characterisation of tractable conjunctive query answering for the bounded degree case.