## Abstract

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 neighborhoods, 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.

Original language | English (US) |
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Pages (from-to) | 69-108 |

Number of pages | 40 |

Journal | Journal of Algorithms |

Volume | 42 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2002 |

## Keywords

- Graph power
- Leaf power
- Phylogenetic tree
- Tree power

## ASJC Scopus subject areas

- Control and Optimization
- Computational Mathematics
- Computational Theory and Mathematics