On the characterization of the source-to-all-terminal diameter-constrained reliability domination

Let G = (V;E) be a digraph with a distinguished set of terminal vertices K V and a vertex s 2 K . We de ne the s;K-diameter of G as the maximum distance between s and any of vertices of K. If the arcs fail randomly and independently with known probabilities (vertices are always operational), the Diameter-constrained s;K-terminal reliability of G, Rs;K(G;D), is de ned as the probability that surviving arcs span a subgraph whose s;K- diameter does not exceed D [5, 11]. A graph invariant called the domination of a graph G was introduced by Satyanarayana and Prabhakar [13] to generate the non-canceling terms of the classical reliability expres- sion, Rs;K(G), based on the same reliability model (i.e. arcs fail randomly and indepen- dently and where nodes are perfect), and de ned as the probability that the surviving arcs span a subgraph of G with unconstrained nite s;K-diameter. This result allowed the generation of rapid algorithms for the computation of Rs;K(G). In this paper we present a characterization of the diameter-constrained s;K-terminal reliability domination of a digraph G = (V;E) with terminal set K = V , and for any diameter bound D, and, as a result, we solve the classical reliability domination, as a speci c case. Moreover we also present a rapid algorithm for the evaluation of Rs;V (G;D).Eje: Teoría (TEOR)Red de Universidades con Carreras en Informática (RedUNCI

Identifying and Analyzing RNA Pseudoknots based on Graph-theoretical Properties of Dual Graphs: a Partitioning Approach

In this paper we propose the study of properties of RNA secondary structures modeled as dual graphs, by partitioning these graphs into topological components denominated blocks. We give a full characterization of possible topological configurations of these blocks, and, in particular we show that an RNA secondary structure contains a pseudoknot if and only if its corresponding dual graph contains a block having a vertex of degree at least 3. Once a dual graph has been partitioned via computationally-efficient well-known graph-theoretical algorithms, this characterization allow us to identify these sub-topologies and physically isolate pseudoknots from RNA secondary structures and analyze them for specific combinatorial properties (e.g., connectivity)

On the characterization of the domination of a diameter-constrained network reliability model

AbstractLet G=(V,E) be a digraph with a distinguished set of terminal vertices K⊆V and a vertex s∈K. We define the s,K-diameter of G as the maximum distance between s and any of the vertices of K. If the arcs fail randomly and independently with known probabilities (vertices are always operational), the diameter-constrained s,K-terminal reliability of G, Rs,K(G,D), is defined as the probability that surviving arcs span a subgraph whose s,K-diameter does not exceed D.The diameter-constrained network reliability is a special case of coherent system models, where the domination invariant has played an important role, both theoretically and for developing algorithms for reliability computation. In this work, we completely characterize the domination of diameter-constrained network models, giving a simple rule for computing its value: if the digraph either has an irrelevant arc, includes a directed cycle or includes a dipath from s to a node in K longer than D, its domination is 0; otherwise, its domination is -1 to the power |E|-|V|+1. In particular this characterization yields the classical source-to-K-terminal reliability domination obtained by Satyanarayana.Based on these theoretical results, we present an algorithm for computing the reliability

Graph-Theoretic Partitioning of RNAs and Classification of Pseudoknots-II

Dual graphs have been applied to model RNA secondary structures with pseudoknots, or intertwined base pairs. In previous works, a linear-time algorithm was introduced to partition dual graphs into maximally connected components called blocks and determine whether each block contains a pseudoknot or not. As pseudoknots can not be contained into two different blocks, this characterization allow us to efficiently isolate smaller RNA fragments and classify them as pseudoknotted or pseudoknot-free regions, while keeping these sub-structures intact. Moreover we have extended the partitioning algorithm by classifying a pseudoknot as either recursive or non-recursive in order to continue with our research in the development of a library of building blocks for RNA design by fragment assembly. In this paper we present a methodology that uses our previous results and classify pseudoknots into the classical H,K,L, and M types, based upon a novel representation of RNA secondary structures as dual directed graphs (i.e., digraphs). This classification would help the systematic analysis of RNA structure and prediction as for example the implementation of more accurate folding algorithms

On the characterization of the source-to-all-terminal diameter-constrained reliability domination

Let G = (V;E) be a digraph with a distinguished set of terminal vertices K V and a vertex s 2 K . We de ne the s;K-diameter of G as the maximum distance between s and any of vertices of K. If the arcs fail randomly and independently with known probabilities (vertices are always operational), the Diameter-constrained s;K-terminal reliability of G, Rs;K(G;D), is de ned as the probability that surviving arcs span a subgraph whose s;K- diameter does not exceed D [5, 11]. A graph invariant called the domination of a graph G was introduced by Satyanarayana and Prabhakar [13] to generate the non-canceling terms of the classical reliability expres- sion, Rs;K(G), based on the same reliability model (i.e. arcs fail randomly and indepen- dently and where nodes are perfect), and de ned as the probability that the surviving arcs span a subgraph of G with unconstrained nite s;K-diameter. This result allowed the generation of rapid algorithms for the computation of Rs;K(G). In this paper we present a characterization of the diameter-constrained s;K-terminal reliability domination of a digraph G = (V;E) with terminal set K = V , and for any diameter bound D, and, as a result, we solve the classical reliability domination, as a speci c case. Moreover we also present a rapid algorithm for the evaluation of Rs;V (G;D).Eje: Teoría (TEOR)Red de Universidades con Carreras en Informática (RedUNCI

Domination Invariant of a Diameter Constrained Network Reliability Model

Let G=(V,E) be a digraph with a distinguished set of terminal vertices K in V and a vertex s in K. We define the s,K-diameter of G as the maximum distance between s and any of vertices of K. If the arcs fail randomly and independently with known probabilities (vertices are always operational), the Diameter-constrained s,K-terminal reliability of G, R_\{s,K\}(G,D) is defined as the probability that surviving arcs span a subgraph whose s,K-diameter does not exceed D. The Diameter-constrained network reliability is a special case of coherent system models, where the domination invariant has played an important role, both theoretically and for developing algorithms for reliability computation. In this work, we completely characterize the domination of diameter-constrained network models, giving a simple rule for computing its value: if the digraph either has an irrelevant edge, includes a dicycle or includes a dipath from $s$ to a node in K longer than D, its domination is 0; otherwise, its domination is -1 to the power |E|-|V|+1

Properties of a generalized source-to-all-terminal network reliability model with diameter constraints

Given the pervasive nature of computer and communication networks, many paradigms have been used to study their properties and performances. In particular, reliability models based on topological properties can adequately represent the network capacity to survive failures of its components. Classical reliability models are based on the existence of end-to-end paths between network nodes, not taking into account the length of these paths; for many applications, this is inadequate, because the connection will only be established or attain the required quality if the distance between the connecting nodes does not exceed a given value. An alternative topological reliability model is the diameter-constrained reliability of a network; this measure considers not only the underlying topology, but also imposes a bound on the diameter, which is the maximum distance between the nodes of the network. In this work, we study in particular the case where we want to model the connection between a source-vertex s and a set of terminal vertices K (for example, a video multicast application), using a directed graph (digraph) for representing the topology of the network with node set V. If the s,K-diameter is the maximum distance between s and any of vertices of K, the diameter-constrained s,K-terminal reliability of a network G, Rs,K(G,D), is defined as the probability that surviving arcs span a subgraph whose s,K-diameter does not exceed D. One of the tools successfully employed in the study of classical reliability models is the domination of a graph, which was introduced by Satyanarayana and Prabhakar. In this paper we introduce a definition and a full characterization of the domination in the case of the diameter-constrained s,K-terminal reliability when K=V, including the classical source-to-all-terminal reliability domination result as a specific case. Moreover we use these results to present an algorithm for the evaluation of the diameter-constrained s,V-terminal reliability Rs,V(G,D).Reliability Networks Diameter Domination

RNA-As-Graphs Motif Atlas—Dual Graph Library of RNA Modules and Viral Frameshifting-Element Applications

RNA motif classification is important for understanding structure/function connections and building phylogenetic relationships. Using our coarse-grained RNA-As-Graphs (RAG) representations, we identify recurrent dual graph motifs in experimentally solved RNA structures based on an improved search algorithm that finds and ranks independent RNA substructures. Our expanded list of 183 existing dual graph motifs reveals five common motifs found in transfer RNA, riboswitch, and ribosomal 5S RNA components. Moreover, we identify three motifs for available viral frameshifting RNA elements, suggesting a correlation between viral structural complexity and frameshifting efficiency. We further partition the RNA substructures into 1844 distinct submotifs, with pseudoknots and junctions retained intact. Common modules are internal loops and three-way junctions, and three submotifs are associated with riboswitches that bind nucleotides, ions, and signaling molecules. Together, our library of existing RNA motifs and submotifs adds to the growing universe of RNA modules, and provides a resource of structures and substructures for novel RNA design