Locality in distributed graph algorithms
SIAM Journal on Computing
Computing roots of graphs is hard
Discrete Applied Mathematics
Journal of Algorithms
Computing Phylogenetic Roots with Bounded Degrees and Errors
WADS '01 Proceedings of the 7th International Workshop on Algorithms and Data Structures
Phylogenetic k-Root and Steiner k-Root
ISAAC '00 Proceedings of the 11th International Conference on Algorithms and Computation
Computing bounded-degree phylogenetic roots of disconnected graphs
WG'04 Proceedings of the 30th international conference on Graph-Theoretic Concepts in Computer Science
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We extend the well-studied concept of a graph power to that of a k-leaf power G of a tree T: G is formed by creating a node for each leaf in the tree and an edge between a pair of nodes if and only if the associated leaves are connected by a path of length at most k. By discovering hidden combinatorial structure of cliques and neighbourhoods, we have developed polynomial-time algorithms that, for k = 3 and k = 4, identify whether or not a given graph G is a k-leaf power of a tree T, and if so, produce a tree T for which G is a k-leaf power. We believe that our structural results will form the basis of a solution for more general k. The general problem of inferring hidden tree structure on the basis of leaf relationships shows up in several areas of application.