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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 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 : = vc ( G 1 ) + 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 ⊆ coNP / poly . Finally, we study MCIS and its connected variant MCCIS on some special graph classes and with respect to other structural parameters. [ABSTRACT FROM AUTHOR]

In this paper, we study three connection games among the most widely played: havannah , twixt , and slither . We show that determining the outcome of an arbitrary input position is PSPACE -complete in all three cases. Our reductions are based on the popular graph problem generalized geography and on hex itself. We also consider the complexity of generalizations of hex parameterized by the length of the solution and establish that while short generalized hex is W[1] -hard, short hex is FPT . Finally, we prove that the ultra-weak solution to the empty starting position in hex cannot be fully adapted to any of these three games. [ABSTRACT FROM AUTHOR]