Abstract
Connected Vertex Cover is one of the classical problems of computer science, already mentioned in the monograph of Garey and Johnson [15]. Although the optimization and decision variants of finding connected vertex covers of minimum size or weight are well studied, surprisingly there is no work on the enumeration or maximum number of minimal connected vertex covers of a graph. In this paper we show that the maximum number of minimal connected vertex covers of a graph is \(O(1.8668^n)\), and these can be enumerated in time \(O(1.8668^n)\). For graphs of chordality at most 5, we are able to give a better upper bound, and for chordal graphs and distance-hereditary graphs we are able to give tight bounds on the maximum number of minimal connected vertex covers.
The research leading to these results has received funding from the Research Council of Norway and the European Research Council under the European Union’s Seventh Framework Programme (FP/2007–2013)/ERC Grant Agreement no. 267959.
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Notes
- 1.
The computations have been done by computer.
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Golovach, P.A., Heggernes, P., Kratsch, D. (2016). Enumeration and Maximum Number of Minimal Connected Vertex Covers in Graphs. In: Lipták, Z., Smyth, W. (eds) Combinatorial Algorithms. IWOCA 2015. Lecture Notes in Computer Science(), vol 9538. Springer, Cham. https://doi.org/10.1007/978-3-319-29516-9_20
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