The treewidth of 2-section of hypergraphs

Ke Liu,Mei Lu

doi:10.46298/dmtcs.6499

Abstract

Let $H=(V,F)$ be a simple hypergraph without loops. $H$ is called linear if $|f\cap g|\le 1$ for any $f,g\in F$ with $f\not=g$. The $2$-section of $H$, denoted by $[H]_2$, is a graph with $V([H]_2)=V$ and for any $ u,v\in V([H]_2)$, $uv\in E([H]_2)$ if and only if there is $ f\in F$ such that $u,v\in f$. The treewidth of a graph is an important invariant in structural and algorithmic graph theory. In this paper, we consider the treewidth of the $2$-section of a linear hypergraph. We will use the minimum degree, maximum degree, anti-rank and average rank of a linear hypergraph to determine the upper and lower bounds of the treewidth of its $2$-section. Since for any graph $G$, there is a linear hypergraph $H$ such that $[H]_2\cong G$, we provide a method to estimate the bound of treewidth of graph by the parameters of the hypergraph.

Highlights

The treewidth of a graph is an important invariant in structural and algorithmic graph theory
In this paper, we study the treewidth of 2-section of linear hypergraphs
In order to cite the definition of a generalized hypertree decomposition of a hypergraph which was given in [4], we introduce the definition of the 2-section of a hypergraph

Summary

Introduction

The treewidth of a graph is an important invariant in structural and algorithmic graph theory. Given a minimum width tree decomposition of [H]2, replace each vertex with all the hyperedges containing the vertex to obtain a supertree decomposition of H We prove the following lower bound on tw([H]2) in terms of the minimum degree δ, maximum degree ∆ and average rank l(H) of a linear hypergraph H. Theorem 1.1 Let H be a linear hypergraph with minimum degree δ, maximum degree ∆ and average rank l(H). We consider a minimum width supertree decomposition of H, and replace each bag λt by the vertices that are incident to an hyperedge of λt This creates a tree decomposition of [H]2, where each bag contains at most r(H)stw(H) vertices.

Tree decomposition of 2-section of hypergraphs

Lower bound in terms of average rank

Lower bounds in terms of anti-rank

Upper bound

Conclusion