Finding hypergraph immersion is fixed-parameter tractable

TL;DR

This paper extends the concept of graph immersion to hypergraphs and proves that determining hypergraph immersion is fixed-parameter tractable with a polynomial-time algorithm.

Contribution

It introduces hypergraph immersion, extends existing graph theory concepts, and demonstrates fixed-parameter tractability with a specific polynomial-time algorithm.

Findings

Hypergraph immersion can be decided in polynomial time.

The problem is fixed-parameter tractable with respect to the size of the hypergraph.

A dual hypergraph immersion problem is also characterized.

Abstract

Immersion minor is an important variant of graph minor, defined through an injective mapping from vertices in a smaller graph $H$ to vertices in a larger graph $G$ where adjacent elements of the former are connected in the latter by edge-disjoint paths. Here, we consider the immersion problem in the emerging field of hypergraphs. We first define hypergraph immersion by extending the injective mapping to hypergraphs. We then prove that finding a hypergraph immersion is fixed-parameter tractable, namely, there exists an $O(N^6)$ polynomial-time algorithm to determine whether a fixed hypergraph $H$ can be immersed in a hypergraph $G$ with $N$ vertices. Additionally, we present the dual hypergraph immersion problem and provide further characteristics of the algorithmic complexity.