Hitting and Harvesting Pumpkins

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 covering and packing c-pumpkin-models in a given graph: On the one hand, we provide an FPT algorithm running in time 2 O(k) n O(1) deciding, for any fixed c ≥ 1, whether all c-pumpkin-models can be covered by at most k vertices. This generalizes known single-exponential FPT algorithms for Vertex Cover and Feedback Vertex Set, which correspond to the cases c = 1, 2 respectively. On the other hand, we present a O(log n)-approximation algorithm for both the problems of covering all c-pumpkin-models with a smallest number of vertices, and packing a maximum number of vertex-disjoint c-pumpkin-models.