Binary functions, degeneracy, and alternating dimaps

TL;DR

This paper explores the properties of binary functions, characterising degeneracy and their relation to alternating dimaps, revealing limitations on their binary representations and opening questions for more complex forms.

Contribution

It characterises degenerate elements in binary functions and investigates their connection to Tutte's alternating dimaps, identifying constraints on binary representations.

Findings

Degenerate elements in binary functions are characterised.

Only the simplest alternating dimaps have binary representations of the studied form.

Open question on existence of more sophisticated binary representations for alternating dimaps.

Abstract

This paper continues the study of combinatorial properties of binary functions --- that is, functions $f:2^E\rightarrow\mathbb{C}$ such that $f(\emptyset)=1$, where $E$ is a finite set. Binary functions have previously been shown to admit families of transforms that generalise duality, including a trinity transform, and families of associated minor operations that generalise deletion and contraction, with both these families parameterised by the complex numbers. Binary function representations exist for graphs (via the indicator functions of their cutset spaces) and indeed arbitrary matroids (as shown by the author previously). In this paper, we characterise degenerate elements --- analogues of loops and coloops --- in binary functions, with respect to any pair of minor operations from our complex-parameterised family. We then apply this to study the relationship between binary functions and Tutte's alternating dimaps, which also support a trinity transform and three associated minor operations. It is shown that only the simplest alternating dimaps have binary representations of the form we consider, which seems to be the most direct type of representation. The question of whether there exist other, more sophisticated types of binary function representations for alternating dimaps is left open.