Some novel minimax results for perfect matchings of hexagonal systems

TL;DR

This paper investigates the anti-forcing number of perfect matchings in hexagonal systems, establishing new minimax results and characterizing systems with triphenylene subgraphs based on anti-forcing properties.

Contribution

It extends minimax results for anti-forcing numbers to hexagonal systems and characterizes systems containing triphenylene subgraphs.

Findings

For perfect matchings with maximum anti-forcing number or minus one, the anti-forcing number equals the count of M-alternating hexagons.

Hexagonal systems with triphenylene subgraphs have anti-forcing numbers equal to the number of M-alternating hexagons for all perfect matchings.

The paper generalizes known results from plane bipartite graphs to hexagonal systems.

Abstract

The anti-forcing number of a perfect matching $M$ of a graph $G$ is the minimum number of edges of $G$ whose deletion results in a subgraph with a unique perfect matching $M$, denoted by $af(G,M)$. When $G$ is a plane bipartite graph, Lei et al. established a minimax result: For any perfect matching $M$ of $G$, $af(G,M)$ equals the maximum number of $M$-alternating cycles of $G$ where any two either are disjoint or intersect only at edges in $M$; For a hexagonal system, the maximum anti-forcing number equals the fries number. In this paper we show that for every perfect matching $M$ of a hexagonal system $H$ with the maximum anti-forcing number or minus one, $af(H,M)$ equals the number of $M$-alternating hexagons of $H$. Further we show that a hexagonal system $H$ has a triphenylene as nice subgraph if and only $af(H,M)$ always equals the number of $M$-alternating hexagons of $H$ for every perfect matching $M$ of $H$.