An induced matching in a graph is a set of edges whose endpoints induce a $1$-regular subgraph. Gupta et al. (2012,\cite{ Gupta}) showed that every $n$-vertex graph has at most $10^{ \frac{ n}{ 5}}\approx 1.5849^n$ maximal induced matchings, which is attained by the disjoint union of copies of the complete graph $K_5$. In this paper, we show that the maximum number of maximal and maximum induced matchings in a connected graph of order $n$ is \begin{ align*} \begin{ cases} { n\choose 2} &~ { \rm if}~ 1\leq n\le 8; \\ { { \lfloor \frac{ n}{ 2} \rfloor}\choose 2}\cdot { { \lceil \frac{ n}{ 2} \rceil}\choose 2} -(\lfloor \frac{ n}{ 2} \rfloor-1)\cdot (\lceil \frac{ n}{ 2} \rceil-1)+1 &~ { \rm if}~ 9\leq n\le 13; \\ 10^{ \frac{ n-1}{ 5}}+\frac{ n+144}{ 30}\cdot 6^{ \frac{ n-6}{ 5}} &~ { \rm if}~ 14\leq n\le 30;\\ 10^{ \frac{ n-1}{ 5}}+\frac{ n-1}{ 5}\cdot 6^{ \frac{ n-6}{ 5}} & ~ { \rm if}~ n\geq 31, \\ \end{ cases} \end{ align*} and also show that this bound is tight. This result implies that we can enumerate all maximal induced matchings of an $n$-vertex connected graph in time $O(1.5849^n)$. Moreover, our result provides an estimate on the number of maximal dissociation sets of an $n$-vertex connected graph.
Categories: math.CO