The parameterized complexity of k-edge induced subgraphs

We prove that finding a k-edge induced subgraph is fixed-parameter tractable, thereby answering an open problem of Leizhen Cai. Our algorithm is based on several combinatorial observations, Gauss' famous Eureka theorem, and a generalization of the well-known fpt-algorithm for the model-checking problem for first-order logic on graphs with locally bounded tree-width due to Frick and Grohe. On the other hand, we show that two natural counting versions of the problem are hard. Hence, the k-edge induced subgraph problem is one of the very few known examples in parameterized complexity that are easy for decision while hard for counting.