Parameterized Complexity of the Smallest Degree-Constrained Subgraph Problem

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

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