[Paper Review] An infinite family of excluded minors for strong base-orderability

Joseph E. Bonin, Thomas J. Savitsky|arXiv (Cornell University)|Jul 20, 2015

Advanced Graph Theory Research13 references4 citations

TL;DR

This paper constructs an infinite family of excluded minors for strong base-orderability by generalizing Ingleton's example, proving that infinitely many of his identified matroids are excluded minors for both base-orderability and gammoids. It introduces k-base-orderability as a hierarchy between base-orderability and strong base-orderability, showing that k-base-orderable matroids form a complete class and that the constructed matroids are (k−1)-base-orderable but not k-base-orderable, establishing their role as excluded minors for strong base-orderability.

ABSTRACT

We discuss a conjecture of Ingleton on excluded minors for base-orderability, and, extending a result he stated, we prove that infinitely many of the matroids that he identified are excluded minors for base-orderability, as well as for the class of gammoids. We prove that a paving matroid is base-orderable if and only if it has no minor that is isomorphic to the cycle matroid of the complete graph on four vertices. For each k that is at least 2, we define the property of k-base-orderability, which lies strictly between base-orderability and strong base-orderability, and we show that k-base-orderable matroids form what Ingleton called a complete class. By generalizing an example of Ingleton, we construct a set of matroids, each of which is an excluded minor for k-base-orderability, but is (k-1)-base-orderable; the union of these sets, over all k, is an infinite set of base-orderable excluded minors for strong base-orderability.

Motivation & Objective

• To resolve Ingleton's conjecture on excluded minors for base-orderability by constructing an infinite family of such minors.
• To define and analyze the hierarchy of k-base-orderable matroids, lying strictly between base-orderability and strong base-orderability.
• To prove that paving matroids are base-orderable if and only if they have no M(K₄)-minor.
• To show that the constructed matroids are excluded minors for both base-orderability and the class of gammoids.
• To establish that the class of k-base-orderable matroids forms a complete class under Ingleton's definition.

Proposed method

• Generalizing Ingleton's example to construct a family of matroids Mβ with specific cyclic flats and circuit-hyperplanes.
• Defining k-base-orderability as a basis-exchange property where exchange holds for all subsets of size at most k.
• Using duals and single-element contractions to show that Mβ and Mβ* are excluded minors for k-base-orderability and strong base-orderability.
• Proving that single-element contractions of Mβ are transversal and hence gammoids, using the Mason-Ingleton condition.
• Analyzing the structure of bases and cyclic flats to verify that Mβ is (k−1)-base-orderable but not k-base-orderable.
• Establishing that the union of these excluded minors over all k ≥ 2 forms an infinite set of excluded minors for strong base-orderability.

Experimental results

Research questions

• RQ1 Are infinitely many of the matroids Ingleton identified in his 1975 paper excluded minors for base-orderability and strong base-orderability?
• RQ2 Can a hierarchy of k-base-orderable matroids be defined such that each level is strictly stronger than the previous and forms a complete class?
• RQ3 Is the class of paving matroids base-orderable if and only if it excludes the M(K₄) minor?
• RQ4 Do the constructed matroids Mβ serve as excluded minors for k-base-orderability while remaining (k−1)-base-orderable?
• RQ5 Are the single-element deletions of Mβ also gammoids, or do they require a different characterization?

Key findings

• An infinite family of excluded minors for strong base-orderability is constructed, each of which is base-orderable but not strongly base-orderable.
• For each k ≥ 2, the matroids Mβ are excluded minors for k-base-orderability but are (k−1)-base-orderable, establishing a strict hierarchy.
• The class of k-base-orderable matroids is proven to be a complete class under Ingleton’s definition, closed under minors, duals, direct sums, truncations, and induction by directed graphs.
• A paving matroid is base-orderable if and only if it has no M(K₄)-minor, providing a complete characterization for this class.
• The dual matroid Mβ* is also an excluded minor for k-base-orderability and strong base-orderability, and is (k−1)-base-orderable.
• The number of non-isomorphic matroids Mβ for a given k is h² when k = 2h+1 and (h−1)² + h when k = 2h, showing a precise count based on k's parity.