Ptolemaic Graphs and Interval Graphs Are Leaf Powers

Brandstädt, Andreas; Hundt, Christian

doi:10.1007/978-3-540-78773-0_42

Cited by 25 publications

(27 citation statements)

References 22 publications

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“…The converse implication is not true as mentioned in [4] (based on [3,13]): There are strongly chordal graphs which are not a k-leaf power for any k ≥ 2.…”

Section: Propositionmentioning

confidence: 83%

“…See [4][5][6][7][8][10][11][12]15,17,18,25,[22][23][24]29] for recent work on k-leaf powers and their variants (including characterisations of 3-leaf powers [5,17,29] as well as of 4-leaf powers [10,29] and a linear time recognition of 5-leaf powers [15]). For k ≥ 6, no characterisation of k-leaf powers and no efficient recognition is known.…”

Section: Introductionmentioning

confidence: 99%

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Characterising (k,ℓ)-leaf powers

Brandstädt

Wagner

2010

Discrete Applied Mathematics

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a b s t r a c tWe say that, for k ≥ 2 and > k, a tree T with distance function d T (x, y) is a (k, )-leaf root of a finite simple graph G = (V , E) if V is the set of leaves of T , for all edges xy ∈ E, d T (x, y) ≤ k, and for all non-edges xy ∈ E, d T (x, y) ≥ . A graph is a (k, )-leaf power if it has a (k, )-leaf root. This new notion modifies the concept of k-leaf powers (which are, in our terminology, the (k, k + 1)-leaf powers) introduced and studied by Nishimura, Ragde and Thilikos; k-leaf powers are motivated by the search for underlying phylogenetic trees. Recently, a lot of work has been done on k-leaf powers and roots as well as on their variants phylogenetic roots and Steiner roots. Many problems, however, remain open.We give the structural characterisations of (k, )-leaf powers, for some k and , which also imply an efficient recognition of these classes, and in this way we improve and extend a recent paper by Kennedy, Lin and Yan on strictly chordal graphs; one of our motivations for studying (k, )-leaf powers is the fact that strictly chordal graphs are precisely the (4, 6)-leaf powers.

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“…The converse implication is not true as mentioned in [4] (based on [3,13]): There are strongly chordal graphs which are not a k-leaf power for any k ≥ 2.…”

Section: Propositionmentioning

confidence: 83%

Section: Introductionmentioning

confidence: 99%

Characterising (k,ℓ)-leaf powers

Brandstädt

Wagner

2010

Discrete Applied Mathematics

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“…Therefore, we focus on the CLP4 EDGE DELETION case in this section. 3 Note that graphs that have 3-leaf roots have a characterization similar to that of Theorem 3.1: they are graphs that are chordal and contain none of the induced subgraphs "bull", "dart", and "gem" [13]. Therefore, the basic idea for CLP3 EDGE DELETION as well as for CLP4 EDGE DELETION is to use the forbidden subgraph characterization in a depth-bounded search tree algorithm: find a forbidden subgraph, and recursively branch into several cases according to the possible edge deletions that destroy the forbidden subgraph.…”

Section: Fixed-parameter Tractability Of Clp4mentioning

confidence: 97%

“…1 k-LEAF POWER is solvable in linear time for k ≤ 5 [4,6,8], and k-PHYLOGENETIC ROOT is solvable in polynomial time for k ≤ 4 [33,28]. The complexities of both recognition problems are open for k ≥ 6 and k ≥ 5, respectively, although it is known that every so-called strictly chordal graph is a k-leaf power for every k [24] (see also [3] for similar results on further graph classes), and 5-PHYLOGENETIC ROOT can be solved in cubic time on strictly chordal graphs [23].…”

Section: Introductionmentioning

confidence: 97%

Closest 4-leaf power is fixed-parameter tractable

Dom¹,

Guo²,

Hüffner³

et al. 2008

Discrete Applied Mathematics

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The NP-complete CLOSEST 4-LEAF 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 4-leaf power. Herein, a 4-leaf power is a graph that can be constructed by considering an unrooted tree-the 4-leaf root-with leaves one-to-one 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 2-LEAF POWER and CLOSEST 3-LEAF POWER, we give the first algorithmic result for CLOSEST 4-LEAF POWER, showing that CLOSEST 4-LEAF POWER is fixed-parameter tractable with respect to the parameter r .

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“…Despite considerable effort, for k ≥ 6, no non-trivial characterization and no efficient recognition of k-leaf powers is known. See [5][6][7]10,29] for more information on leaf powers and in particular, for new characterizations of 3-and 4-leaf powers as well as of distance-hereditary 5-leaf powers.…”

Section: Introductionmentioning

confidence: 99%

Simplicial powers of graphs

Brandstädt

Lê

2009

Theoretical Computer Science

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a b s t r a c tIn a graph, a vertex is simplicial if its neighborhood is a clique. For an integer k ≥ 1, athe set of all simplicial vertices of H, and for all distinct vertices x and y in V G , xy ∈ E G if and only if the distance in H between x and y is at most k. This concept generalizes k-leaf powers introduced by Nishimura, Ragde and Thilikos which were motivated by the search for underlying phylogenetic trees; k-leaf powers are the k-simplicial powers of trees.Recently, a lot of work has been done on k-leaf powers and their roots as well as on their variants phylogenetic roots and Steiner roots. For k ≤ 5, k-leaf powers can be recognized in linear time, and for k ≤ 4, structural characterizations are known. For k ≥ 6, the recognition and characterization problems of k-leaf powers are still open. Since trees and block graphs (i.e., connected graphs whose blocks are cliques) have very similar metric properties, it is natural to study k-simplicial powers of block graphs. We show that leaf powers of trees and simplicial powers of block graphs are closely related, and we study simplicial powers of other graph classes containing all trees such as ptolemaic graphs and strongly chordal graphs.

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Ptolemaic Graphs and Interval Graphs Are Leaf Powers

Cited by 25 publications

References 22 publications

Characterising (k,ℓ)-leaf powers

Characterising (k,ℓ)-leaf powers

Closest 4-leaf power is fixed-parameter tractable

Simplicial powers of graphs

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