Finding hypergraph immersion is fixed-parameter tractable

TL;DR

It is proved 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.

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.

Figures (10)

• Figure 1:
• Figure 2:
• Figure 3:
• Figure 5: Converting hypergraph $G$ to ordinary graphs. The ordinary graph $G_{M}'$ is the $M$-generalised factor graph of $G$, and $G_{M,L}"$ is the densified version of $G_{M}'$.
• Figure 6: Hypergraph $H$ and its factor graph $H'$. While $H$ can be immersed in $G$, the factor graph $H'$ may not be embedded in $G_{M}'$.

Theorems & Definitions (42)

• Definition 2.1: immersion in ordinary graphs robertson-seymour_rs10
• Definition 2.2
• Remark 2.3
• Definition 2.4: embedding in ordinary graphs graph-theor-appl
• Definition 2.5
• Remark 2.6
• Theorem 2.7: time complexity of embedding in ordinary graphs embed-fix-param-tract_gkmw11
• Definition 3.1
• Definition 3.2
• Remark 3.3