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# Parameterized Complexity of the Smallest Degree Constraint Subgraph Problem

1 MASCOTTE - Algorithms, simulation, combinatorics and optimization for telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : 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.
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https://hal.inria.fr/inria-00157970
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Submitted on : Tuesday, July 3, 2007 - 5:37:23 PM
Last modification on : Friday, February 4, 2022 - 3:11:29 AM
Long-term archiving on: : Friday, September 24, 2010 - 11:00:09 AM

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• HAL Id : inria-00157970, version 4

### Citation

Omid Amini, Ignasi Sau Valls, Saket Saurabh. Parameterized Complexity of the Smallest Degree Constraint Subgraph Problem. [Research Report] RR-6237, INRIA. 2007, pp.20. ⟨inria-00157970v4⟩

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