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A linear-time kernelization for the rooted \(k\)-leaf outbranching problem. (English) Zbl 1317.05072
Summary: In the rooted \(k\)-leaf outbranching problem, a digraph \(G = (V, E)\), a vertex \(r\) of \(G\), and an integer \(k\) are given, and the goal is to find an \(r\)-rooted spanning outtree of \(G\) with \(\geq k\) leaves (a subtree of \(G\) with vertex set \(V\), all edges directed away from \(r\), and \(\geq k\) leaves). We present a linear-time algorithm that computes a problem kernel with \(O(k^6)\) vertices and \(O(k^7)\) edges for the rooted \(k\)-leaf outbranching problem. By combining the new result with a result of J. Daligault and S. Thomassé [Lect. Notes Comput. Sci. 5917, 86–97 (2009; Zbl 1273.68162)], a kernel with a quadratic number of vertices and edges can be found on \(n\)-vertex \(m\)-edge digraphs in time \(O(n + m + k^{14})\).

MSC:
05C20 Directed graphs (digraphs), tournaments
05C05 Trees
05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
68Q25 Analysis of algorithms and problem complexity
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