A network flow approach to a common generalization of Clar and Fries numbers

TL;DR

This paper introduces a unified network flow framework for generalizing Clar and Fries numbers in bipartite plane graphs and extends it to directed graphs, providing new algorithms and theoretical insights.

Contribution

It presents the first network flow based algorithm for optimal Fries structures and generalizes Clar and Fries numbers through a unified approach with min-max theorems.

Findings

Developed a linear programming duality for the generalized parameters

Provided a strongly polynomial network flow algorithm for the problem

Extended the framework to non-planar directed graphs

Abstract

Clar number and Fries number are two thoroughly investigated parameters of plane graphs emerging from mathematical chemistry to measure stability of organic molecules. We consider first a common generalization of these two concepts for bipartite plane graphs, and then extend it to a framework on general (not necessarily planar) directed graphs. The corresponding optimization problem can be transformed into a maximum weight feasible tension problem which is the linear programming dual of a minimum cost network flow (or circulation) problem. Therefore the approach gives rise to a min-max theorem and to a strongly polynomial algorithm that relies exclusively on standard network flow subroutines. In particular, we give the first network flow based algorithm for an optimal Fries structure and its variants.