129 research outputs found
Computing Optimal Leaf Roots of Chordal Cographs in Linear Time
A graph G is a k-leaf power, for an integer k >= 2, if there is a tree T with
leaf set V(G) such that, for all vertices x, y in V(G), the edge xy exists in G
if and only if the distance between x and y in T is at most k. Such a tree T is
called a k-leaf root of G. The computational problem of constructing a k-leaf
root for a given graph G and an integer k, if any, is motivated by the
challenge from computational biology to reconstruct phylogenetic trees. For
fixed k, Lafond [SODA 2022] recently solved this problem in polynomial time.
In this paper, we propose to study optimal leaf roots of graphs G, that is,
the k-leaf roots of G with minimum k value. Thus, all k'-leaf roots of G
satisfy k <= k'. In terms of computational biology, seeking optimal leaf roots
is more justified as they yield more probable phylogenetic trees. Lafond's
result does not imply polynomial-time computability of optimal leaf roots,
because, even for optimal k-leaf roots, k may (exponentially) depend on the
size of G. This paper presents a linear-time construction of optimal leaf roots
for chordal cographs (also known as trivially perfect graphs). Additionally, it
highlights the importance of the parity of the parameter k and provides a
deeper insight into the differences between optimal k-leaf roots of even versus
odd k.
Keywords: k-leaf power, k-leaf root, optimal k-leaf root, trivially perfect
leaf power, chordal cographComment: 22 pages, 2 figures, full version of the FCT 2023 pape
Parameterized Leaf Power Recognition via Embedding into Graph Products
The k-leaf power graph G of a tree T is a graph whose vertices are the leaves of T and whose edges connect pairs of leaves at unweighted distance at most k in T. Recognition of the k-leaf power graphs for k >= 6 is still an open problem. In this paper, we provide an algorithm for this problem for sparse leaf power graphs. Our result shows that the problem of recognizing these graphs is fixed-parameter tractable when parameterized both by k and by the degeneracy of the given graph. To prove this, we describe how to embed the leaf root of a leaf power graph into a product of the graph with a cycle graph. We bound the treewidth of the resulting product in terms of k and the degeneracy of G. As a result, we can use methods based on monadic second-order logic (MSO_2) to recognize the existence of a leaf power as a subgraph of the product graph
Treewidth distance on phylogenetic trees
In this article we study the treewidth of the display graph, an auxiliary graph structure obtained from the fusion of phylogenetic (i.e., evolutionary) trees at their leaves. Earlier work has shown that the treewidth of the display graph is bounded if the trees are in some formal sense topologically similar. Here we further expand upon this relationship. We analyse a number of reduction rules, commonly used in the phylogenetics literature to obtain fixed parameter tractable algorithms. In some cases (the subtree reduction) the reduction rules behave similarly with respect to treewidth, while others (the cluster reduction) behave very differently, and the behaviour of the chain reduction is particularly intriguing because of its link with graph separators and forbidden minors. We also show that the gap between treewidth and Tree Bisection and Reconnect (TBR) distance can be infinitely large, and that unlike, for example, planar graphs the treewidth of the display graph can be as much as linear in its number of vertices. A number of other auxiliary results are given. We conclude with a discussion and list a number of open problems
Polynomial kernels for 3-leaf power graph modification problems
A graph G=(V,E) is a 3-leaf power iff there exists a tree T whose leaves are
V and such that (u,v) is an edge iff u and v are at distance at most 3 in T.
The 3-leaf power graph edge modification problems, i.e. edition (also known as
the closest 3-leaf power), completion and edge-deletion, 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, Huffner and
Niedermeier (2005).Comment: Submitte
Graph Powers: Hardness Results, Good Characterizations and Efficient Algorithms
Given a graph H = (V_H,E_H) and a positive integer k, the k-th power of H, written H^k, is the graph obtained from H by adding edges between any pair of vertices at distance at most k in H; formally, H^k = (V_H, {xy | 1 <= d_H (x, y) <= k}). A graph G is the k-th power of a graph H if G = H^k, and in this case, H is a k-th root of G. Our investigations deal with the computational complexity of recognizing k-th powers of general graphs as well as restricted graphs. This work provides new NP-completeness results, good characterizations and efficient algorithms for graph powers
Pairwise Compatibility Graphs: A Survey
International audienceA graph is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree and two nonnegative real numbers and such that each leaf of is a node of and there is an edge if and only if , where is the sum of weights of the edges on the unique path from to in . In this article, we survey the state of the art concerning this class of graphs and some of its subclasses
Local Certification of Some Geometric Intersection Graph Classes
In the context of distributed certification, the recognition of graph classes
has started to be intensively studied. For instance, different results related
to the recognition of planar, bounded tree-width and -minor free graphs have
been recently obtained. The goal of the present work is to design compact
certificates for the local recognition of relevant geometric intersection graph
classes, namely interval, chordal, circular arc, trapezoid and permutation.
More precisely, we give proof labeling schemes recognizing each of these
classes with logarithmic-sized certificates. We also provide tight logarithmic
lower bounds on the size of the certificates on the proof labeling schemes for
the recognition of any of the aforementioned geometric intersection graph
classes
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