On the Complexity of Various Parameterizations of Common Induced Subgraph Isomorphism

Maximum Common Induced Subgraph (henceforth MCIS) is among the most studied classical \({\mathsf {NP}}\)-hard problems. MCIS remains \({\mathsf {NP}}\)-hard on many graph classes including bipartite graphs, planar graphs and k-trees. Little is known, however, about the parameterized complexity of the problem. When parameterized by the vertex cover number of the input graphs, the problem was recently shown to be fixed-parameter tractable. Capitalizing on this result, we show that the problem does not have a polynomial kernel when parameterized by vertex cover unless \({\mathsf {NP}}\subseteq \mathsf {coNP}/poly\). We also show that Maximum Common Connected Induced Subgraph (MCCIS), which is a variant where the solution must be connected, is also fixed-parameter tractable when parameterized by the vertex cover number of input graphs. Both problems are shown to be \({\mathsf {W[1]}}\)-complete on bipartite graphs and graphs of girth five and, unless \({\mathsf {P}}= {\mathsf {NP}}\), they do not belong to the class \({\mathsf {XP}}\) when parameterized by a bound on the size of the minimum feedback vertex sets of the input graphs, that is solving them in polynomial time is very unlikely when this parameter is a constant.