14,435 research outputs found
Equivalence of the filament and overlap graphs of subtrees of limited trees
The overlap graphs of subtrees of a tree are equivalent to subtree filament
graphs, the overlap graphs of subtrees of a star are cocomparability graphs,
and the overlap graphs of subtrees of a caterpillar are interval filament
graphs. In this paper, we show the equivalence of many more classes of subtree
overlap and subtree filament graphs, and equate them to classes of complements
of cochordal-mixed graphs. Our results generalize the previously known results
mentioned above
Graph kernels based on tree patterns for molecules
Motivated by chemical applications, we revisit and extend a family of
positive definite kernels for graphs based on the detection of common subtrees,
initially proposed by Ramon et al. (2003). We propose new kernels with a
parameter to control the complexity of the subtrees used as features to
represent the graphs. This parameter allows to smoothly interpolate between
classical graph kernels based on the count of common walks, on the one hand,
and kernels that emphasize the detection of large common subtrees, on the other
hand. We also propose two modular extensions to this formulation. The first
extension increases the number of subtrees that define the feature space, and
the second one removes noisy features from the graph representations. We
validate experimentally these new kernels on binary classification tasks
consisting in discriminating toxic and non-toxic molecules with support vector
machines
Random subtrees of complete graphs
We study the asymptotic behavior of four statistics associated with subtrees
of complete graphs: the uniform probability that a random subtree is a
spanning tree of , the weighted probability (where the probability a
subtree is chosen is proportional to the number of edges in the subtree) that a
random subtree spans and the two expectations associated with these two
probabilities. We find and both approach ,
while both expectations approach the size of a spanning tree, i.e., a random
subtree of has approximately edges
Partitions and Coverings of Trees by Bounded-Degree Subtrees
This paper addresses the following questions for a given tree and integer
: (1) What is the minimum number of degree- subtrees that partition
? (2) What is the minimum number of degree- subtrees that cover
? We answer the first question by providing an explicit formula for the
minimum number of subtrees, and we describe a linear time algorithm that finds
the corresponding partition. For the second question, we present a polynomial
time algorithm that computes a minimum covering. We then establish a tight
bound on the number of subtrees in coverings of trees with given maximum degree
and pathwidth. Our results show that pathwidth is the right parameter to
consider when studying coverings of trees by degree-3 subtrees. We briefly
consider coverings of general graphs by connected subgraphs of bounded degree
Recognising the overlap graphs of subtrees of restricted trees is hard
The overlap graphs of subtrees in a tree (SOGs) generalise many other graphs classes with set representation characterisations. The complexity of recognising SOGs is open. The complexities of recognising many subclasses of SOGs are known. Weconsider several subclasses of SOGs by restricting the underlying tree. For a fixed integer , we consider:\begin{my_itemize} \item The overlap graphs of subtrees in a tree where that tree has leaves
\item The overlap graphs of subtrees in trees that can be derived from a given input tree by subdivision and have at least 3 leaves \item The overlap and intersection graphs of paths in a tree where that tree has maximum degree \end{my_itemize}
We show that the recognition problems of these classes are NP-complete. For all other parameters we get circle graphs, well known to be polynomially recognizable
Generation of random chordal graphs using subtrees of a tree
Chordal graphs form one of the most studied graph classes. Several graph problems that are NP-hard in general become solvable in polynomial time on chordal graphs, whereas many others remain NP-hard. For a large group of problems among the latter, approximation algorithms, parameterized algorithms, and algorithms with moderately exponential or sub-exponential running time have been designed. Chordal graphs have also gained increasing interest during the recent years in the area of enumeration algorithms. Being able to test these algorithms on instances of chordal graphs is crucial for understanding the concepts of tractability of hard problems on graph classes. Unfortunately, only few published papers give algorithms for generating chordal graphs. Even in these papers, only very few methods aim for generating a large variety of chordal graphs. Surprisingly, none of these methods is directly based on the “intersection of subtrees of a tree” characterization of chordal graphs. In this paper, we give an algorithm for generating chordal graphs, based on the characterization that a graph is chordal if and only if it is the intersection graph of subtrees of a tree. Upon generating a random host tree, we give and test various methods that generate subtrees of the host tree. We compare our methods to one another and to existing ones for generating chordal graphs. Our experiments show that one of our methods is able to generate the largest variety of chordal graphs in terms of maximal clique sizes. Moreover, two of our subtree generation methods result in an overall complexity of our generation algorithm that is the best possible time complexity for a method generating the entire node set of subtrees in an “intersection of subtrees of a tree” representation. The instances corresponding to the results presented in this paper, and also a set of relatively small-sized instances are made available online.publishedVersio
Recognising the overlap graphs of subtrees of restricted trees is hard
The overlap graphs of subtrees in a tree (SOGs) generalise many other graphs classes with set representation characterisations. The complexity of recognising SOGs is open. The complexities of recognising many subclasses of SOGs are known. Weconsider several subclasses of SOGs by restricting the underlying tree. For a fixed integer , we consider:\begin{my_itemize} \item The overlap graphs of subtrees in a tree where that tree has leaves
\item The overlap graphs of subtrees in trees that can be derived from a given input tree by subdivision and have at least 3 leaves \item The overlap and intersection graphs of paths in a tree where that tree has maximum degree \end{my_itemize}
We show that the recognition problems of these classes are NP-complete. For all other parameters we get circle graphs, well known to be polynomially recognizable
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