Abstract
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 (MSMD d ). 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 MSMS d 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.
This work has been partially supported by European project IST FET AEOLUS, PACA region of France, Ministerio de Educación y Ciencia of Spain, European Regional Development Fund under project TEC2005-03575, Catalan Research Council under project 2005SGR00256, and COST action 293 GRAAL, and has been done in the context of the crc Corso with France Telecom.
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References
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Amini, O., Sau, I., Saurabh, S. (2008). Parameterized Complexity of the Smallest Degree-Constrained Subgraph Problem. In: Grohe, M., Niedermeier, R. (eds) Parameterized and Exact Computation. IWPEC 2008. Lecture Notes in Computer Science, vol 5018. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79723-4_4
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DOI: https://doi.org/10.1007/978-3-540-79723-4_4
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