Basis Number and Pathwidth

TL;DR

This paper establishes bounds relating the basis number of a graph to its pathwidth and path decompositions, showing polynomial bounds for graphs with certain minor exclusions.

Contribution

It proves that the basis number is at most four times the pathwidth and provides bounds for graphs with bounded adhesion path decompositions, extending to minor-free graphs.

Findings

Basis number is at most four times the pathwidth.

Basis number of K_t-minor-free graphs is polynomial in t.

Graphs with bounded adhesion path decompositions have bounded basis number.

Abstract

We prove two results relating the basis number of a graph $G$ to path decompositions of $G$. Our first result shows that the basis number of a graph is at most four times its pathwidth. Our second result shows that, if a graph $G$ has a path decomposition with adhesions of size at most $k$ in which the graph induced by each bag has basis number at most $b$, then $G$ has basis number at most $b+O(k\log^2 k)$. The first result, combined with recent work of Geniet and Giocanti shows that the basis number of a graph is bounded by a polynomial function of its treewidth. The second result (also combined with the work of Geniet and Giocanti) shows that every $K_t$-minor-free graph has a basis number bounded by a polynomial function of $t$.