On the Optimal Representation of Algebraic Expressions of Fibonacci Graphs

TL;DR

This paper explores methods to simplify algebraic expressions of Fibonacci graphs, aiming to find their shortest representations, and introduces an optimal decomposition approach with an open conjecture.

Contribution

It proposes an optimal decomposition method for Fibonacci graph expressions and presents an open problem regarding its conjectured optimality.

Findings

Proposed a new decomposition method for Fibonacci graph expressions

Identified the problem of finding shortest algebraic representations

Presented an open conjecture on the optimality of the method

Abstract

The paper investigates relationship between algebraic expressions and graphs. We consider a digraph called a Fibonacci graph which gives a generic example of non-series-parallel graphs. Our intention in this paper is to simplify the expressions of Fibonacci graphs and eventually find their shortest representations. With that end in view, we describe the optimal decomposition method for generating Fibonacci graph expressions that is conjectured to provide these representations. Proof (or disproof) of this conjecture is presented as an open problem.