[Paper Review] Minors for alternating dimaps

Graham Farr|arXiv (Cornell University)|Nov 12, 2013

Advanced Combinatorial Mathematics21 references3 citations

TL;DR

This paper develops a minor theory for alternating dimaps—orientably embedded digraphs with alternating in- and out-edges at each vertex—introducing three minor operations related by Tutte's triality. It establishes excluded minor characterizations for bounded genus, proves non-commutativity of minor operations, and defines simple and extended Tutte invariants, showing they generalize the Tutte polynomial for planar graphs.

ABSTRACT

We develop a theory of minors for alternating dimaps --- orientably embedded digraphs where, at each vertex, the incident edges (taken in the order given by the embedding) are directed alternately into, and out of, the vertex. We show that they are related by the triality relation of Tutte. They do not commute in general, though do in many circumstances, and we characterise the situations where they do. The relationship with triality is reminiscent of similar relationships for binary functions, due to the author, so we characterise those alternating dimaps which correspond to binary functions. We give a characterisation of alternating dimaps of at most a given genus, using a finite set of excluded minors. We also use the minor operations to define simple Tutte invariants for alternating dimaps and characterise them. We establish a connection with the Tutte polynomial, and pose the problem of characterising universal Tutte-like invariants for alternating dimaps based on these minor operations.

Motivation & Objective

• To develop a comprehensive minor theory for alternating dimaps, extending classical graph minor concepts to orientable embedded digraphs.
• To investigate the non-commutativity of minor operations and characterize when they do commute.
• To establish a finite excluded minor characterization for alternating dimaps of bounded genus.
• To define and analyze simple and extended Tutte invariants for alternating dimaps, linking them to the classical Tutte polynomial.
• To explore the possibility of universal Tutte-like invariants using the new minor operations and triality structure.

Proposed method

• Introduces three minor operations—1-reduction, ω-reduction, and ω²-reduction—on alternating dimaps, generalizing deletion and contraction.
• Demonstrates that these operations satisfy a triality relation analogous to Tutte’s triality for graphs and binary functions.
• Uses the triality structure to analyze when minor operations commute, identifying conditions under which the operations preserve order independence.
• Applies the minor operations to derive a finite set of excluded minors for alternating dimaps of genus at most g, for any fixed g.
• Defines simple Tutte invariants via recursive relations based on edge types (loops, coloops, etc.), and extended invariants using ordered edge reductions.
• Connects extended Tutte invariants to the Tutte polynomial of planar graphs by showing that invariants on alt_c(H) or alt_a(H) are independent of edge order.

Experimental results

Research questions

• RQ1 How can minor operations be defined and characterized for alternating dimaps, and what is their relationship to triality?
• RQ2 Under what conditions do the three minor operations on alternating dimaps commute?
• RQ3 Can alternating dimaps of bounded genus be characterized by a finite set of excluded minors?
• RQ4 What are the properties and limitations of simple and extended Tutte invariants for alternating dimaps?
• RQ5 To what extent can the Tutte polynomial of a planar graph be recovered as a special case of extended Tutte invariants on alternating dimaps?

Key findings

• The three minor operations on alternating dimaps satisfy a triality relation analogous to that in Tutte’s theory and in the author’s earlier work on binary functions.
• Minor operations on alternating dimaps do not commute in general, but they do commute when the edges involved are not in the same face or star structure.
• For any fixed genus g, the class of alternating dimaps of genus at most g has a finite excluded minor characterization.
• Simple Tutte invariants for alternating dimaps are limited in scope and contain significantly less information than their counterparts in graph or matroid theory.
• Extended Tutte invariants, defined using ordered edge reductions, are richer and include the Tutte polynomial of a planar graph as a special case.
• When the underlying graph H is 4-regular and G = alt_c(H), the extended Tutte invariants are independent of the edge ordering, due to the presence of ω²-loops after each reduction.