8 research outputs found
Exploring structural properties of -trees and block graphs
We present a new characterization of -trees based on their reduced clique
graphs and -line graphs, which are block graphs. We explore structural
properties of these two classes, showing that the number of clique-trees of a
-tree equals the number of spanning trees of the -line graph of
. This relationship allows to present a new approach for determining the
number of spanning trees of any connected block graph. We show that these
results can be accomplished in linear time complexity.Comment: 6 pages, 1 figur
Unique Perfect Phylogeny Characterizations via Uniquely Representable Chordal Graphs
The perfect phylogeny problem is a classic problem in computational biology,
where we seek an unrooted phylogeny that is compatible with a set of
qualitative characters. Such a tree exists precisely when an intersection graph
associated with the character set, called the partition intersection graph, can
be triangulated using a restricted set of fill edges. Semple and Steel used the
partition intersection graph to characterize when a character set has a unique
perfect phylogeny. Bordewich, Huber, and Semple showed how to use the partition
intersection graph to find a maximum compatible set of characters. In this
paper, we build on these results, characterizing when a unique perfect
phylogeny exists for a subset of partial characters. Our characterization is
stated in terms of minimal triangulations of the partition intersection graph
that are uniquely representable, also known as ur-chordal graphs. Our
characterization is motivated by the structure of ur-chordal graphs, and the
fact that the block structure of minimal triangulations is mirrored in the
graph that has been triangulated
Two methods for the generation of chordal graphs
In this paper two methods for automatic generation of connected chordal graphs are proposed: the first one is based on results concerning the dynamic maintainance of chordality under edge insertions; the second is based on expansion/merging of maximal cliques. In both methods, chordality is preserved along the whole generation process
Determining what sets of trees can be the clique trees of a chordal graph
Chordal graphs have characteristic tree representations, the clique trees. The problems of finding one or enumerating them have already been solved in a satisfactory way. In this paper, the following related problem is studied: given a family T of trees, all having the same vertex set V, determine whether there exists a chordal graph whose set of clique trees equals T. For that purpose, we undertake a study of the structural properties, some already known and some new, of the clique trees of a chordal graph and the characteristics of the sets that induce subtrees of every clique tree. Some necessary and sufficient conditions and examples of how they can be applied are found, eventually establishing that a positive or negative answer to the problem can be obtained in polynomial time. If affirmative, a graph whose set of clique trees equals T is also obtained. Finally, all the chordal graphs with set of clique trees equal to T are characterized.Facultad de Ciencias Exacta
Determining what sets of trees can be the clique trees of a chordal graph
Chordal graphs have characteristic tree representations, the clique trees. The problems of finding one or enumerating them have already been solved in a satisfactory way. In this paper, the following related problem is studied: given a family T of trees, all having the same vertex set V, determine whether there exists a chordal graph whose set of clique trees equals T. For that purpose, we undertake a study of the structural properties, some already known and some new, of the clique trees of a chordal graph and the characteristics of the sets that induce subtrees of every clique tree. Some necessary and sufficient conditions and examples of how they can be applied are found, eventually establishing that a positive or negative answer to the problem can be obtained in polynomial time. If affirmative, a graph whose set of clique trees equals T is also obtained. Finally, all the chordal graphs with set of clique trees equal to T are characterized.Facultad de Ciencias Exacta