  In the Maximum Common Induced Subgraph problem (henceforth MCIS), given two
graphs $G_1$ and $G_2$, one looks for a graph with the maximum number of
vertices being both an induced subgraph of $G_1$ and $G_2$. MCIS is among the
most studied classical NP-hard problems. It remains NP-hard on many graph
classes including forests. In this paper, we study the parameterized complexity
of MCIS. As a generalization of \textsc{Clique}, it is W[1]-hard parameterized
by the size of the solution. Being NP-hard even on forests, most structural
parameterizations are intractable. One has to go as far as parameterizing by
the size of the minimum vertex cover to get some tractability. Indeed, when
parameterized by $k := \text{vc}(G_1)+\text{vc}(G_2)$ the sum of the vertex
cover number of the two input graphs, the problem was shown to be
fixed-parameter tractable, with an algorithm running in time $2^{O(k \log k)}$.
We complement this result by showing that, unless the ETH fails, it cannot be
solved in time $2^{o(k \log k)}$. This kind of tight lower bound has been shown
for a few problems and parameters but, to the best of our knowledge, not for
the vertex cover number. We also show that MCIS does not have a polynomial
kernel when parameterized by $k$, unless $NP \subseteq \mathsf{coNP}/poly$.
Finally, we study MCIS and its connected variant MCCIS on some special graph
classes and with respect to other structural parameters.
