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An Euler tour in a hypergraph is a closed walk that traverses each edge ofthe hypergraph exactly once, while an Euler family, first defined by Bahmanianand Sajna, is a family of closed walks that jointly traverse each edge exactlyonce and cannot be concatenated. In this paper, we study the notions of aspanning Euler tour and a spanning Euler family, that is, an Euler tour(family) that also traverses each vertex of the hypergraph at least once. Weexamine necessary and sufficient conditions for a hypergraph to admit aspanning Euler family, most notably, when the hypergraph possesses a vertex cutconsisting of vertices of degree two. Moreover, we characterise hypergraphswith a vertex cut of cardinality at most two that admit a spanning Euler tour(family). This result enables us to reduce the problem of existence of aspanning Euler tour (which is NP-complete), as well as the problem of aspanning Euler family, to smaller hypergraphs.

Spanning Euler tours and spanning Euler families in hypergraphs with particular vertex cuts