Parameterized Complexity of the Smallest Degree-Constrained Subgraph Problem

In this paper we study the problem of finding an induced subgraph of size at most k with minimum degree at least d for a given graph G, from the parameterized complexity perspective. We call this problem Minimum Subgraph of Minimum Degree ≥d (MSMDd). For d = 2 it corresponds to finding a shortest cycle of the graph. Our main motivation to study this problem is its strong relation to Dense k-Subgraph and Traffic Grooming problems. First, we show that MSMSd is fixed-parameter intractable (provided FPT = W [1]) for d ≥ 3 in general graphs, by showing it to be W [1]-hard using a reduction from Multi-Color Clique. In the second part of the paper we provide explicit fixed-parameter tractable (FPT) algorithms for the problem in graphs with bounded local tree-width and graphs with excluded minors, faster than those coming from the meta-theorem of Frick and Grohe [13] about problems definable in first order logic over “locally tree-decomposable structures”. In particular, this implies faster fixed-parameter tractable algorithms in planar graphs, graphs of bounded genus, and graphs with bounded maximum degree.