Parameterized Complexity of the Smallest Degree Constraint Subgraph Problem

In this paper we initiate the study of finding an induced subgraph of size at most $k$ with minimum degree at least $d$. We call this problem {\sc Minimum Subgraph of Minimum Degree $_{\geq d}$ (MSMD$_d$)}. For $d=2$, it corresponds to finding a shortest cycle of the graph. The problem is strongly related to the \textsc{Dense $k$-Subgraph} problem and is of interest in practical applications. We show that the {\sc MSMS}$_d$ is fixed parameter intractable for $d\geq 3$ in general graphs by showing it to be W[1]-hard by a reduction from {\sc Multi-Color Clique}. On the algorithmic side, we show that the problem is fixed parameter tractable in graphs which excluded minors and graphs with bounded local tree-width. In particular, this implies that the problem is fixed parameter tractable in planar graphs, graphs of bounded genus and graphs with bounded maximum degree