Abstract
A k-leaf power of a tree T is a graph G whose vertices are the leaves of T and whose edges connect pairs of leaves whose distance in T is at most k. A graph is a leaf power if it is a k-leaf power for some k. Over 20 years ago, Nishimura et al. [J. Algorithms, 2002] asked if recognition of leaf powers was in P. Recently, Lafond [SODA 2022] showed an XP algorithm when parameterized by k, while leaving the main question open. In this paper, we explore this question from the perspective of two alternative models of leaf powers, showing that both a linear and a star variant of leaf powers can be recognized in polynomial-time.
Omitted proofs and a conclusion can be found in the full version of this paper available on https://arxiv.org/abs/2105.12407.
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Benjamin, B., Høgemo, S., Telle, J.A., Vatshelle, M. (2022). Recognition of Linear and Star Variants of Leaf Powers is in P. In: Bekos, M.A., Kaufmann, M. (eds) Graph-Theoretic Concepts in Computer Science. WG 2022. Lecture Notes in Computer Science, vol 13453. Springer, Cham. https://doi.org/10.1007/978-3-031-15914-5_6
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