Efficient algorithms for domination and Hamilton circuit problems on permutation graphs

The domination and Hamilton circuit problems are of interest both in algorithm design and complexity theory. The domination problem has applications in facility location and the Hamilton circuit problem has applications in routing problems in communications and operations research.The problem of deciding if G has a dominating set of cardinality at most k, and the problem of determining if G has a Hamilton circuit are NP-Complete. Polynomial time algorithms are, however, available for a large number of restricted classes. A motivation for the study of these algorithms is that they not only give insight into the characterization of these classes but also require a variety of algorithmic techniques and data structures. So the search for efficient algorithms, for these problems in many classes still continues.A class of perfect graphs which is practically important and mathematically interesting is the class of permutation graphs. The domination problem is polynomial time solvable on permutation graphs. Algorithms that are already available are of time complexity O(n2) or more, and space complexity O(n2) on these graphs. The Hamilton circuit problem is open for this class.We present a simple O(n) time and O(n) space algorithm for the domination problem on permutation graphs. Unlike the existing algorithms, we use the concept of geometric representation of permutation graphs. Further, exploiting this geometric notion, we develop an O(n2) time and O(n) space algorithm for the Hamilton circuit problem.

Anti-apoptotic Protein BCL2 Down-regulates DNA End Joining in Cancer Cells

Cancer cells are often associated with secondary chromosomal rearrangements, such as deletions, inversions, and translocations, which could be the consequence of unrepaired/misrepaired DNA double strand breaks (DSBs). Nonhomologous DNA end joining is one of the most common pathways to repair DSBs in higher eukaryotes. By using oligomeric DNA substrates mimicking various endogenous DSBs in a cell-free system, we studied end joining (EJ) in different cancer cell lines. We found that the efficiency of EJ varies among cancer cells; however, there was no remarkable difference in the mechanism and expression of EJ proteins. Interestingly, cancer cells with lower levels of EJ possessed elevated expression of BCL2 and vice versa. Removal of BCL2 by immunoprecipitation or protein fractionation led to elevated EJ. More importantly, we show that overexpression of BCL2 or the addition of purified BCL2 led to the down-regulation of EJ. Further, we found that BCL2 interacts with KU proteins both in vitro and in vivo. Hence, our results suggest that EJ in cancer cells could be negatively regulated by the anti-apoptotic protein, BCL2, and this may contribute toward increased chromosomal abnormalities in cancer.

Criticality of the exponential rate of decay for the largest nearest-neighbor link in random geometric graphs

Let n points be placed independently in d-dimensional space according to the density f(x) = A(d)e(-lambda parallel to x parallel to alpha), lambda, alpha > 0, x is an element of R-d, d >= 2. Let d(n) be the longest edge length of the nearest-neighbor graph on these points. We show that (lambda(-1) log n)(1-1/alpha) d(n) - b(n) converges weakly to the Gumbel distribution, where b(n) similar to ((d - 1)/lambda alpha) log log n. We also prove the following strong law for the normalized nearest-neighbor distance (d) over tilde (n) = (lambda(-1) log n)(1-1/alpha) d(n)/log log n: (d - 1)/alpha lambda <= lim inf(n ->infinity) (d) over tilde (n) <= lim sup(n ->infinity) (d) over tilde (n) <= d/alpha lambda almost surely. Thus, the exponential rate of decay alpha = 1 is critical, in the sense that, for alpha > 1, d(n) -> 0, whereas, for alpha <= 1, d(n) -> infinity almost surely as n -> infinity.

Over-expression and purification strategies for recombinant multi-protein oligomers: A case study of Mycobacterium tuberculosis sigma/anti-sigma factor protein complexes

The function of a protein in a cell often involves coordinated interactions with one or several regulatory partners. It is thus imperative to characterize a protein both in isolation as well as in the context of its complex with an interacting partner. High resolution structural information determined by X-ray crystallography and Nuclear Magnetic Resonance offer the best route to characterize protein complexes. These techniques, however, require highly purified and homogenous protein samples at high concentration. This requirement often presents a major hurdle for structural studies. Here we present a strategy based on co-expression and co-purification to obtain recombinant multi-protein complexes in the quantity and concentration range that can enable hitherto intractable structural projects. The feasibility of this strategy was examined using the sigma factor/anti-sigma factor protein complexes from Mycobacterium tuberculosis. The approach was successful across a wide range of sigma factors and their cognate interacting partners. It thus appears likely that the analysis of these complexes based on variations in expression constructs and procedures for the purification and characterization of these recombinant protein samples would be widely applicable for other multi-protein systems. (C) 2010 Elsevier Inc. All rights reserved.

Tree 3-spanners in 2-sep chordal graphs: Characterization and algorithms

A spanning tree T of a graph G is said to be a tree t-spanner if the distance between any two vertices in T is at most t times their distance in G. A graph that has a tree t-spanner is called a tree t-spanner admissible graph. The problem of deciding whether a graph is tree t-spanner admissible is NP-complete for any fixed t >= 4 and is linearly solvable for t <= 2. The case t = 3 still remains open. A chordal graph is called a 2-sep chordal graph if all of its minimal a - b vertex separators for every pair of non-adjacent vertices a and b are of size two. It is known that not all 2-sep chordal graphs admit tree 3-spanners This paper presents a structural characterization and a linear time recognition algorithm of tree 3-spanner admissible 2-sep chordal graphs. Finally, a linear time algorithm to construct a tree 3-spanner of a tree 3-spanner admissible 2-sep chordal graph is proposed. (C) 2010 Elsevier B.V. All rights reserved.

Cubicity of Interval Graphs and the Claw Number

Let G(V, E) be a simple, undirected graph where V is the set of vertices and E is the set of edges. A b-dimensional cube is a Cartesian product l(1) x l(2) x ... x l(b), where each l(i) is a closed interval of unit length on the real line. The cub/city of G, denoted by cub(G), is the minimum positive integer b such that the vertices in G can be mapped to axis parallel b-dimensional cubes in such a way that two vertices are adjacent in G if and only if their assigned cubes intersect. An interval graph is a graph that can be represented as the intersection of intervals on the real line-i.e. the vertices of an interval graph can be mapped to intervals on the real line such that two vertices are adjacent if and only if their corresponding intervals overlap. Suppose S(m) denotes a star graph on m+1 nodes. We define claw number psi(G) of the graph to be the largest positive integer m such that S(m) is an induced subgraph of G. It can be easily shown that the cubicity of any graph is at least log(2) psi(G)]. In this article, we show that for an interval graph G log(2) psi(G)-]<= cub(G)<=log(2) psi(G)]+2. It is not clear whether the upper bound of log(2) psi(G)]+2 is tight: till now we are unable to find any interval graph with cub(G)> (log(2)psi(G)]. We also show that for an interval graph G, cub(G) <= log(2) alpha], where alpha is the independence number of G. Therefore, in the special case of psi(G)=alpha, cub(G) is exactly log(2) alpha(2)]. The concept of cubicity can be generalized by considering boxes instead of cubes. A b-dimensional box is a Cartesian product l(1) x l(2) x ... x l(b), where each I is a closed interval on the real line. The boxicity of a graph, denoted box(G), is the minimum k such that G is the intersection graph of k-dimensional boxes. It is clear that box(G)<= cub(G). From the above result, it follows that for any graph G, cub(G) <= box(G)log(2) alpha]. (C) 2010 Wiley Periodicals, Inc. J Graph Theory 65: 323-333, 2010

Analysing local algorithms in location-aware quasi-unit-disk graphs

A local algorithm with local horizon r is a distributed algorithm that runs in r synchronous communication rounds; here r is a constant that does not depend on the size of the network. As a consequence, the output of a node in a local algorithm only depends on the input within r hops from the node. We give tight bounds on the local horizon for a class of local algorithms for combinatorial problems on unit-disk graphs (UDGs). Most of our bounds are due to a refined analysis of existing approaches, while others are obtained by suggesting new algorithms. The algorithms we consider are based on network decompositions guided by a rectangular tiling of the plane. The algorithms are applied to matching, independent set, graph colouring, vertex cover, and dominating set. We also study local algorithms on quasi-UDGs, which are a popular generalisation of UDGs, aimed at more realistic modelling of communication between the network nodes. Analysing the local algorithms on quasi-UDGs allows one to assume that the nodes know their coordinates only approximately, up to an additive error. Despite the localisation error, the quality of the solution to problems on quasi-UDGs remains the same as for the case of UDGs with perfect location awareness. We analyse the increase in the local horizon that comes along with moving from UDGs to quasi-UDGs.

Communication pathways from the anti-codon region to the aminoacylation site in Methionyl tRNA Synthetase: Molecular dynamics simulations and the structure network analysis

The enzymes of the family of tRNA synthetases perform their functions with high precision by synchronously recognizing the anticodon region and the aminoacylation region, which are separated by ?70 in space. This precision in function is brought about by establishing good communication paths between the two regions. We have modeled the structure of the complex consisting of Escherichia coli methionyl-tRNA synthetase (MetRS), tRNA, and the activated methionine. Molecular dynamics simulations have been performed on the modeled structure to obtain the equilibrated structure of the complex and the cross-correlations between the residues in MetRS have been evaluated. Furthermore, the network analysis on these simulated structures has been carried out to elucidate the paths of communication between the activation site and the anticodon recognition site. This study has provided the detailed paths of communication, which are consistent with experimental results. Similar studies also have been carried out on the complexes (MetRS + activated methonine) and (MetRS + tRNA) along with ligand-free native enzyme. A comparison of the paths derived from the four simulations clearly has shown that the communication path is strongly correlated and unique to the enzyme complex, which is bound to both the tRNA and the activated methionine. The details of the method of our investigation and the biological implications of the results are presented in this article. The method developed here also could be used to investigate any protein system where the function takes place through long-distance communication.

Metabolome based reaction graphs of M.tuberculosis and M.leprae: A comparative network analysis

Background. Several types of networks, such as transcriptional, metabolic or protein-protein interaction networks of various organisms have been constructed, that have provided a variety of insights into metabolism and regulation. Here, we seek to exploit the reaction-based networks of three organisms for comparative genomics. We use concepts from spectral graph theory to systematically determine how differences in basic metabolism of organisms are reflected at the systems level and in the overall topological structures of their metabolic networks. Methodology/Principal Findings. Metabolome-based reaction networks of Mycobacterium tuberculosis, Mycobacterium leprae and Escherichia coli have been constructed based on the KEGG LIGAND database, followed by graph spectral analysis of the network to identify hubs as well as the sub-clustering of reactions. The shortest and alternate paths in the reaction networks have also been examined. Sub-cluster profiling demonstrates that reactions of the mycolic acid pathway in mycobacteria form a tightly connected sub-cluster. Identification of hubs reveals reactions involving glutamate to be central to mycobacterial metabolism, and pyruvate to be at the centre of the E. coli metabolome. The analysis of shortest paths between reactions has revealed several paths that are shorter than well established pathways. Conclusions. We conclude that severe downsizing of the leprae genome has not significantly altered the global structure of its reaction network but has reduced the total number of alternate paths between its reactions while keeping the shortest paths between them intact. The hubs in the mycobacterial networks that are absent in the human metabolome can be explored as potential drug targets. This work demonstrates the usefulness of constructing metabolome based networks of organisms and the feasibility of their analyses through graph spectral methods. The insights obtained from such studies provide a broad overview of the similarities and differences between organisms, taking comparative genomics studies to a higher dimension.

Non-contractible non-edges in 2-connected graphs

Any pair of non-adjacent vertices forms a non-edge in a graph. Contraction of a non-edge merges two non-adjacent vertices into a single vertex such that the edges incident on the non-adjacent vertices are now incident on the merged vertex. In this paper, we consider simple connected graphs, hence parallel edges are removed after contraction. The minimum number of nodes whose removal disconnects the graph is the connectivity of the graph. We say a graph is k-connected, if its connectivity is k. A non-edge in a k-connected graph is contractible if its contraction does not result in a graph of lower connectivity. Otherwise the non-edge is non-contractible. We focus our study on non-contractible non-edges in 2-connected graphs. We show that cycles are the only 2-connected graphs in which every non-edge is non-contractible. (C) 2010 Elsevier B.V. All rights reserved.