Kernelization for Finding Lineal Topologies (Depth-First Spanning Trees) with Many or Few Leaves
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For a given graph G, a depth-first search (DFS) tree T of G is an r-rooted spanning tree such that every edge of G is either an edge of T or is between a descendant and an ancestor in T. A graph G together with a DFS tree is called a lineal topology 𝒯 = (G, r, T). Sam et al. (2023) initiated study of the parameterized complexity of the Min-LLT and Max-LLT problems which ask, given a graph G and an integer k≥ 0, whether G has a DFS tree with at most k and at least k leaves, respectively. Particularly, they showed that for the dual parameterization, where the tasks are to find DFS trees with at least n-k and at most n-k leaves, respectively, these problems are fixed-parameter tractable when parameterized by k. However, the proofs were based on Courcelle's theorem, thereby making the running times a tower of exponentials. We prove that both problems admit polynomial kernels with (k^3) vertices. In particular, this implies FPT algorithms running in k^(k)· n^O(1) time. We achieve these results by making use of a (k)-sized vertex cover structure associated with each problem. This also allows us to demonstrate polynomial kernels for Min-LLT and Max-LLT for the structural parameterization by the vertex cover number.
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