Results 1 
4 of
4
On Leaf Powers
"... For an integer k, a tree T is a kleaf root of a finite simple undirected graph G = (V, E) if the set of leaves of T is the vertex set V of G and for any two vertices x, y ∈ V, x ̸ = y, xy ∈ E if and only if the distance of x and y in T is at most k. Then graph G is a kleaf power if it has a kleaf ..."
Abstract

Cited by 15 (2 self)
 Add to MetaCart
For an integer k, a tree T is a kleaf root of a finite simple undirected graph G = (V, E) if the set of leaves of T is the vertex set V of G and for any two vertices x, y ∈ V, x ̸ = y, xy ∈ E if and only if the distance of x and y in T is at most k. Then graph G is a kleaf power if it has a kleaf root. G is a leaf power if it is a kleaf power for some k. This notion was introduced and studied by Nishimura, Ragde and Thilikos; it has its background and motivation in computational biology and phylogeny. In this survey, we describe recent results on leaf powers, variants and generalizations. We discuss the relationship between leaf powers and strongly chordal graphs as well as fixed tolerance NeST graphs, describe some subclasses of leaf powers, give the complete inclusion structure of kleaf power classes, and describe various characterizations of 3and 4leaf powers, as well as of distancehereditary 5leaf powers. Finally we discuss two variants of the notion of kleaf power such as (k, ℓ)leaf powers and exact leaf powers, and we generalize leaf powers (of trees) to simplicial powers of graphs. Most of the presented results are part of joint work, mostly with Van Bang Le and Peter Wagner, but also with Christian Hundt, Federico Mancini, R. Sritharan, and Dieter Rautenbach.
Polynomial kernels for 3leaf power graph modification problems
 In International Workshop on Combinatorial Algorithm (IWOCA), Lecture Notes in Computer Science
, 2009
"... A graph G = (V, E) is a 3leaf power iff there exists a tree T whose leaves are V and such that (u, v) ∈ E iff u and v are at distance at most 3 in T. The 3leaf power graph edge modification problems, i.e. edition (also known as the closest 3leaf power), completion and edgedeletion, are FTP when ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
(Show Context)
A graph G = (V, E) is a 3leaf power iff there exists a tree T whose leaves are V and such that (u, v) ∈ E iff u and v are at distance at most 3 in T. The 3leaf power graph edge modification problems, i.e. edition (also known as the closest 3leaf power), completion and edgedeletion, are FTP when parameterized by the size of the edge set modification. However polynomial kernel was known for none of these three problems. For each of them, we provide cubic kernels that can be computed in linear time for each of these problems. We thereby answer an open problem first mentioned by Dom, Guo, Hüffner and Niedermeier [6]. Research supported by the ANR anrblan014806project ”Graph Decomposition on Algorithm (GRAAL)”
Closest 4Leaf Power is FixedParameter Tractable
, 2008
"... The NPcomplete Closest 4Leaf Power problem asks, given an undirected graph, whether it can be modified by at most r edge insertions or deletions such that it becomes a 4leaf power. Herein, a 4leaf power is a graph that can be constructed by considering an unrooted tree—the 4leaf root—with leave ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
The NPcomplete Closest 4Leaf Power problem asks, given an undirected graph, whether it can be modified by at most r edge insertions or deletions such that it becomes a 4leaf power. Herein, a 4leaf power is a graph that can be constructed by considering an unrooted tree—the 4leaf root—with leaves onetoone labeled by the graph vertices, where we connect two graph vertices by an edge iff their corresponding leaves are at distance at most 4 in the tree. Complementing previous work on Closest 2Leaf Power and Closest 3Leaf Power, we give the first algorithmic result for Closest 4Leaf Power, showing that Closest 4Leaf Power is fixedparameter tractable with respect to the parameter r.
Strictly chordal graphs and . . .
, 2005
"... A phylogeny is the evolutionary history for a set of evolutionarily related species. The development of hereditary trees, or phylogenetic trees, is an important research subject in computational biology. One development approach, motivated by graph theory, constructs a relationship graph based on ev ..."
Abstract
 Add to MetaCart
A phylogeny is the evolutionary history for a set of evolutionarily related species. The development of hereditary trees, or phylogenetic trees, is an important research subject in computational biology. One development approach, motivated by graph theory, constructs a relationship graph based on evolutionary proximity of pairs of species. Associated with this approach is the kth phylogenetic root construction problem: given a relationship graph, construct a phylogenetic tree such that the leaves of the tree correspond to the species and are within distance k if adjacent in the relationship graph. In this thesis, we give a polynomial time algorithm to solve this problem for strictly chordal graphs, a particular subclass of chordal graphs. During the construction of a solution, we examine the problem for tree chordal graphs, and establish new properties for strictly chordal graphs.