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

TL;DR

This paper introduces a new concept of q-linear paths in hypergraphs of rank k, focusing on the case q=k-2, and provides polynomial-time algorithms for related connectivity and game decision problems.

Contribution

It defines and analyzes (k-2)-linear paths in hypergraphs, offering a polynomial-time method to compute connected components and applications to game and path problems.

Findings

Connected components can be computed in polynomial time.

Deciding the Maker-Breaker game winner on rank 3 hypergraphs is polynomial.

Certain NP-complete problems become tractable through hypergraph line graph recognition.

Abstract

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.