Authors: Florian Galliot ^1,2,3,4; Sylvain Gravier ^1,2,3,4; Isabelle Sivignon ^5,2,6,3,4

• 1 Institut Fourier
• 2 Université Grenoble Alpes
• 3 Centre National de la Recherche Scientifique
• 4 Maths à Modeler
• 5 GIPSA Pôle Géométrie, Apprentissage, Information et Algorithmes
• 6 Institut polytechnique de Grenoble - Grenoble Institute of Technology

We define a q-linear path in a hypergraph H as a sequence (e_1,...,e_L) of edges of H such that |e_i ∩ e_i+1 | ∈ [[1, q]] and e_i ∩ e_j = ∅ if |i − j| > 1. In this paper, we study the connected components associated to these paths when q = k − 2 where k is the rank of H. If k = 3 then q = 1 which coincides with the well-known notion of linear path or loose path. We describe the structure of the connected components, using an algorithmic proof which shows that the connected components can be computed in polynomial time. We then mention two consequences of our algorithmic result. The first one is that deciding the winner of the Maker-Breaker game on a hypergraph of rank 3 can be done in polynomial time. The second one is that tractable cases for the NP-complete problem of "Paths Avoiding Forbidden Pairs" in a graph can be deduced from the recognition of a special type of line graph of a hypergraph.