Maximal and maximum induced matchings in connected graphs

Abstract

An induced matching is defined as a set of edges whose end-vertices induce a subgraph that is 1-regular. Building upon the work of Gupta et al. (2012) [11] and Basavaraju et al. (2016) [1], who determined the maximum number of maximal induced matchings in general and triangle-free graphs respectively, this paper extends their findings to connected graphs with n vertices. We establish a tight upper bound on the number of maximal and maximum induced matchings, as detailed below:$\{\begin{array}{cc}\left(\begin{array}{c}n\\ 2\end{array}\right)& \mathrm{if}1\le n\le 8;\\ \left(\begin{array}{c}\lfloor \frac{n}{2}\rfloor \\ 2\end{array}\right)\cdot \left(\begin{array}{c}\lceil \frac{n}{2}\rceil \\ 2\end{array}\right)-(\lfloor \frac{n}{2}\rfloor -1)\cdot (\lceil \frac{n}{2}\rceil -1)+1& \mathrm{if}9\le n\le 13;\\ {10}^{\frac{n-1}{5}}+\frac{n+144}{30}\cdot 6^{\frac{n-6}{5}}& \mathrm{if}14\le n\le 30;\\ {10}^{\frac{n-1}{5}}+\frac{n-1}{5}\cdot 6^{\frac{n-6}{5}}& \mathrm{if}n\ge 31.\end{array}$ This result not only provides a theoretical upper bound but also implies a practical algorithmic application: enumerating all maximal induced matchings of an n-vertex connected graph in time $O\left({1.5849}^n\right)$. Additionally, our work offers an estimate for the number of maximal dissociation sets in connected graphs with n vertices.