Anti-forcing numbers of perfect matchings of graphs

TL;DR

This paper introduces the concept of anti-forcing numbers for perfect matchings in graphs, establishing key minimax and extremal properties, especially in plane bipartite and hexagonal graphs, with implications for molecular graph analysis.

Contribution

It defines the anti-forcing number for perfect matchings and proves a minimax theorem relating it to alternating cycles, also linking maximum anti-forcing numbers to the Fries number in hexagonal systems.

Findings

Anti-forcing number equals the maximal number of disjoint or intersecting M-alternating cycles.

Maximum anti-forcing number of a hexagonal system equals its Fries number.

The Fries number is bounded between the Clar number and twice the Clar number.

Abstract

We define the anti-forcing number of a perfect matching $M$ of a graph $G$ as the minimal number of edges of $G$ whose deletion results in a subgraph with a unique perfect matching $M$, denoted by $af(G,M)$. The anti-forcing number of a graph proposed by Vuki\v{c}evi\'{c} and Trinajsti\'c in Kekul\'e structures of molecular graphs is in fact the minimum anti-forcing number of perfect matchings. For plane bipartite graph $G$ with a perfect matching $M$, we obtain a minimax result: $af(G,M)$ equals the maximal number of $M$-alternating cycles of $G$ where any two either are disjoint or intersect only at edges in $M$. For a hexagonal system $H$, we show that the maximum anti-forcing number of $H$ equals the Fries number of $H$. As a consequence, we have that the Fries number of $H$ is between the Clar number of $H$ and twice. Further, some extremal graphs are discussed.