Forcing and anti-forcing polynomials of a polyomino graph

TL;DR

This paper derives the forcing and anti-forcing polynomials for polyomino graphs, analyzes their spectra, and explores asymptotic behaviors of these graph invariants.

Contribution

It introduces explicit formulas for the forcing and anti-forcing polynomials of polyomino graphs and studies their spectral and asymptotic properties.

Findings

Explicit forcing and anti-forcing polynomials obtained

Spectra of the polyomino graphs determined

Asymptotic behaviors of degree of freedom and sum of anti-forcing numbers revealed

Abstract

The forcing number of a perfect matching $M$ in a graph $G$ is the smallest number of edges inside $M$ that can not be contained in other perfect matchings. The anti-forcing number of $M$ is the smallest number of edges outside $M$ whose removal results in a subgraph with a single perfect matching, that is $M$. Recently, in order to investigate the distributions of forcing numbers and anti-forcing numbers, the forcing polynomial and anti-forcing polynomial were proposed, respectively. In this work, the forcing and anti-forcing polynomials of a polyomino graph are obtained. As consequences, the forcing and anti-forcing spectra of this polyomino graph are determined, and the asymptotic behaviors on the degree of freedom and the sum of all anti-forcing numbers are revealed, respectively.