Reducing to independent set structure: the case of k-internal spanning tree

The k-INTERNAL SPANNING TREE problem asks whether a certain graph G has a spanning tree with at least k internal vertices. Basing our work on the results presented in [Prieto and Sloper 2003], we show that there exists a set of reduction rules that modify an arbitrary spanning tree of a graph into a spanning tree with no induced edges between the leaves. Thus, the rules either produce a tree with many internal vertices, effectively deciding the problem, or they identify a large independent set, the leaves, in the graph. Having a large independent set is beneficial, because then the graph allows both 'crown decompositions' and path decompositions. We show how this crown decomposition can be used to obtain a O(k2) kernel for the k-INTERNAL SPANNING TREE problem, improving on the O(k3) kernel presented in [Prieto and Sloper 2003].