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

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.