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, 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.

翻译：我们将超图$H$中的一条$q$-线性路径定义为边序列$(e_1,\ldots,e_L)$，满足$|e_i \cap e_{i+1}| \in [\![1,q]\!]$且当$|i-j|>1$时$e_i \cap e_j=\varnothing$。本文研究当$q=k-2$（其中$k$为$H$的秩）时这些路径所关联的连通分量。当$k=3$时$q=1$，这对应于熟知的线性路径或松散路径概念。我们通过算法性证明描述了连通分量的结构，表明这些连通分量可在多项式时间内计算。随后我们指出该算法结果的两个推论：其一是在秩为3的超图上判定Maker-Breaker博弈的胜者可在多项式时间内完成；其二是图论中NP完全问题"避免禁止对的路径"的可解情形可由超图某类特殊线图的识别推导得出。