Abstract
The c-pumpkin is the graph with two vertices linked by c ≥ 1 parallel edges. A c-pumpkin-model in a graph G is a pair {A, B} of disjoint subsets of vertices of G, each inducing a connected subgraph of G, such that there are at least c edges in G between A and B. We focus on hitting and packing c-pumpkin-models in a given graph: On the one hand, we provide an FPT algorithm running in time \(2^{\mathcal{O}(k)} n^{\mathcal{O}(1)}\) deciding, for any fixed c ≥ 1, whether all c-pumpkin-models can be hit by at most k vertices. This generalizes the single-exponential FPT algorithms for Vertex Cover and Feedback Vertex Set, which correspond to the cases c = 1,2 respectively. For this, we use a combination of iterative compression and a kernelization-like technique. On the other hand, we present an \(\mathcal{O}(\log n)\)-approximation algorithm for both the problems of hitting all c-pumpkin-models with a smallest number of vertices, and packing a maximum number of vertex-disjoint c-pumpkin-models. Our main ingredient here is a combinatorial lemma saying that any properly reduced n-vertex graph has a c-pumpkin-model of size at most f(c) logn, for a function f depending only on c.
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Joret, G., Paul, C., Sau, I., Saurabh, S., Thomassé, S. (2011). Hitting and Harvesting Pumpkins. In: Demetrescu, C., Halldórsson, M.M. (eds) Algorithms – ESA 2011. ESA 2011. Lecture Notes in Computer Science, vol 6942. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23719-5_34
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DOI: https://doi.org/10.1007/978-3-642-23719-5_34
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