Understanding the structural repertoire of RNA is crucial for RNA genomics research. Yet current methods for ®nding novel RNAs are limited to small or known RNA families. To expand known RNA structural motifs, we develop a two-dimensional graphical representation approach for describing and estimating the size of RNA's secondary structural repertoire, including naturally occurring and other possible RNA motifs. We employ tree graphs to describe RNA tree motifs and more general (dual) graphs to describe both RNA tree and pseudoknot motifs. Our estimates of RNA's structural space are vastly smaller than the nucleotide sequence space, suggesting a new avenue for ®nding novel RNAs. Speci®cally our survey shows that known RNA trees and pseudoknots represent only a small subset of all possible motifs, implying that some of the `missing ' motifs may represent novel RNAs. To help pinpoint RNA-like motifs, we show that the motifs of existing functional RNAs are clustered in a narrow range of topological characteristics. We also illustrate the applications of our approach to the design of novel RNAs and automated comparison of RNA structures; we report several occurrences of RNA motifs within larger RNAs. Thus, our graph theory approach to RNA structures has implications for RNA genomics, structure analysis and design.

A minimum spanning tree (MST) with a small diameter is required in numerous practical situations. It is needed, for example, in distributed mutual exclusion algorithms in order to minimize the number of messages communicated among processors per critical section. Understanding the behavior of tree diameter is useful, for example, in determining an upper bound on the expected number of links between two arbitrary documents on the World Wide Web. The DiameterConstrained MST (DCMST) problem can be stated as follows: given an undirected, edge-weighted graph G with n nodes and a positive integer k, find a spanning tree with the smallest weight among all spanning trees of G which contain no path with more than k edges. This problem is known to be NP-complete, for all values of k; 4 k #n - 2). In this paper, we investigate the behavior of the diameter of MST in randomly-weighted complete graphs (in Erds-Rnyi sense) and explore heuristics for the DCMST problem. For the case when the diameter bound k is small---independent of n, we present a one-time-tree-construction (OTTC) algorithm. It constructs a DCMST in a modified greedy fashion, employing a heuristic for selecting an edge to be added to the tree at each stage of the tree construction. This algorithm is fast and easily parallelizable. We also present a second algorithm that outperforms OTTC for larger values of k. It starts by generating an unconstrained MST and iteratively refines it by replacing edges, one by one, in the middle of long paths in the spanning tree until there is no path left with more than k edges. As expected, the performance of this heuristic is determined by the diameter of the unconstrained MST in the given graph. We discuss convergence, relative merits, and implementation of t...

In numerous practical applications, it is necessary to find the smallest possible tree with a bounded diameter. A diameter-constrained minimum spanning tree (DCMST) of a given undirected, edge-weighted graph, G, is the smallest-weight spanning tree of all spanning trees of G which contain no path with more than k edges, where k is a given positive integer. The problem of finding a DCMST is NP-complete for all values of k; 4 k (n -- 2), except when all edge-weights are identical. A DCMST is essential for the efficiency of various distributed mutual exclusion algorithms, where it can minimize the number of messages communicated among processors per critical section. It is also useful in linear lightwave networks, where it can minimize interference in the network by limiting the traffic in the network lines. Another practical application requiring a DCMST arises in data compression, where some algorithms compress a file utilizing a tree data-structure, and decompress a path in the tree to access a record. A DCMST helps such algorithms to be fast without sacrificing a lot of storage space. We present a survey of the literature on the DCMST problem, study the expected diameter of a random labeled tree, and present five new polynomial-time algorithms for an approximate DCMST. One of our new algorithms constructs an approximate DCMST in a modified greedy fashion, employing a heuristic for selecting an edge to be added to iii the tree in each stage of the construction. Three other new algorithms start with an unconstrained minimum spanning tree, and iteratively refine it into an approximate DCMST. We also present an algorithm designed for the special case when the diameter is required to be no more than 4. Such a diameter-4 tree is also used for evaluating the quality of o...

Abstract. The polytope structure of the associahedron is decomposed into two categories, types and classes. The classification of types is related to integer partitions, whereas the classes present a new combinatorial problem. We solve this, generalizing the work of [25], and incorporate the results into properties of the moduli space Mn 0 (R) studied in [8]. Connections are discussed with relation to classic combinatorial problems as well as to other sciences. 1.1. The Riemann moduli space M n g 1. The Real Moduli Space of surfaces of genus g with n marked points has become a central object in mathematical physics. Introduced in Algebraic Geometry, there is a natural compactification M n g of these spaces; their importance was emphasized by Grothendieck in his famous Esquisse d’un programme [12]. The special case when g = 0, the space M n 0 (C) of n punctures on the sphere CP1, is a building block leading to higher genera. They play a crucial role in the theory

The goal of this paper is to present a panorama of the fundamental properties of cycle index series and asymmetry index series within enumerative combinatorics, as well as a few concrete applications. A given structure is said to be asymmetric if its automorphism group reduces to the identity. We introduce an asymmetry indicator series \Gamma F (x 1 ; x 2 ; x 3 ; : : : ) by means of which we study the correspondence F ! F in connection with the various operations existing in the theory of species of structures. It is shown that all these operations are automatically computable but this aspect is not developed in the summary. 1. Species and Asymmetry Index Series Given any finite set U , let us denote by A[U ] the set of all rooted trees having U as underlying set of vertices. Clearly, every bijection fi : U ! V between finite sets induces another bijection which we denote by A[fi] : A[U ] ! A[V ] and call the transportation of rooted trees along fi (we replace each vertex u in a by t...

In this report, we discuss the labeled and unlabeled enumeration of graphs, connected graphs, trees, eulerian graphs etc. Digraphs and multigraphs are not discussed much. In the labeled case, the use of exponential generating functions is discussed. For the unlabeled case, we only consider applications of Burnside's theorem and Polya's theorem. The use of pair group, product and cartesian product of groups is also discussed.

© 2004 Fera et al; licensee BioMed Central Ltd. This is an Open Access article: verbatim copying and redistribution of this article are permitted in all media for any purpose, provided this notice is preserved along with the article's original URL. Background: The proliferation of structural and functional studies of RNA has revealed an increasing range of RNA's structural repertoire. Toward the objective of systematic cataloguing of RNA's structural repertoire, we have recently described the basis of a graphical approach for organizing RNA secondary structures, including existing and hypothetical motifs. Description: We now present an RNA motif database based on graph theory, termed RAG for RNA-As-Graphs, to catalogue and rank all theoretically possible, including existing, candidate and hypothetical, RNA secondary motifs. The candidate motifs are predicted using a clustering algorithm that classifies RNA graphs into RNA-like and non-RNA groups. All RNA motifs are filed according to their graph vertex number (RNA length) and ranked by topological complexity. Conclusions: RAG's quantitative cataloguing allows facile retrieval of all classes of RNA secondary motifs, assists identification of structural and functional properties of user-supplied RNA sequences, and helps stimulate the search for novel RNAs based on predicted candidate motifs.