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We define a $q$-linear path in a hypergraph $H$ as a sequence$(e_1,\ldots,e_L)$ of edges of $H$ such that $|e_i \cap e_{i+1}| \in[\![1,q]\!]$ and $e_i \cap e_j=\varnothing$ if $|i-j|>1$. In this paper, westudy 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-knownnotion of linear path or loose path. We describe the structure of the connectedcomponents, using an algorithmic proof which shows that the connectedcomponents can be computed in polynomial time. We then mention two consequencesof our algorithmic result. The first one is that deciding the winner of theMaker-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 "PathsAvoiding Forbidden Pairs" in a graph can be deduced from the recognition of aspecial type of line graph of a hypergraph.

(k-2)-linear connected components in hypergraphs of rank k