On the forcing spectrum of generalized Petersen graphs P(n,2)

TL;DR

This paper analyzes the forcing spectrum of perfect matchings in generalized Petersen graphs P(n,2), classifying matchings and describing the spectrum as a union of two specific integer intervals based on n.

Contribution

It classifies perfect matchings of P(n,2) into two types and explicitly determines their forcing spectrum as a union of two intervals for large n.

Findings

For n ≥ 34, forcing spectrum is two unions of integer intervals.

Spectrum depends on n modulo 7 through δ(n).

Provides explicit formulas for the spectrum intervals.

Abstract

The forcing number of a perfect matching $M$ of a graph $G$ is the smallest cardinality of subsets of $M$ that are contained in no other perfect matchings of $G$. The forcing spectrum of $G$ is the collection of forcing numbers of all perfect matchings of $G$. In this paper, we classify the perfect matchings of a generalized Petersen graph $P(n,2)$ in two types, and show that the forcing spectrum is the union of two integer intervals. For $n\ge 34$, it is $\left[\lceil \frac { n }{ 12 } \rceil+1,\lceil \frac { n+3 }{ 7 } \rceil +\delta (n)\right]\cup \left[\lceil \frac { n+2 }{ 6 } \rceil,\lceil \frac { n }{ 4 } \rceil\right]$, where $\delta (n)=1$ if $n\equiv 3$ (mod 7), and $\delta (n)=0$ otherwise.