On the complexity of various parameterizations of common induced subgraph isomorphism

Faisal N. Abu-Khzam, Édouard Bonnet, Florian Sikora
2017 Theoretical Computer Science  
Maximum Common Induced Subgraph (henceforth MCIS) is among the most studied classical NP-hard problems. MCIS remains 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
more » ... en parameterized by vertex cover unless NP ⊆ 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 W[1]-complete on bipartite graphs and graphs of girth five and, unless P = NP, they do not to belong to the class 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.
doi:10.1016/j.tcs.2017.07.010 fatcat:dabbehi67bcwlmjt2ygu67b5dy