Independent set

The cardinality of a largest independent set of G, denoted by α(G), is called the independence number of G. The independent domination number i(G) of a graph G is the cardinality of a smallest independent dominating set of G. We introduce the concept of the common independence number of a graph G, denoted by αc(G), as the greatest integer r such that every vertex of G belongs to some independent subset X of VG with |X|≥r. The common independence number αc(G) of G is the limit of symmetry in G with respect to the fact that each vertex of G belongs to an independent set of cardinality αc(G) in G, and there are vertices in G that do not belong to any larger independent set in G. For any graph G, the relations between above parameters are given by the chain of inequalities i(G)≤αc(G)≤α(G). In this paper, we characterize the trees T for which i(T)=αc(T), and the block graphs G for which αc(G)=α(G).

A perfect independent set I of a graph G is defined to be an independent set with the property that any vertex not in I has at least two neighbors in I. For a nonnegative integer k, a subset I of the vertex set V (G) of a graph G is said to be k-independent, if I is independent and every independent subset where G[I, V (G) -I] is the bipartite graph obtained from G by deleting all edges which are not incident with vertices of I. It is easy to see that a set I is 0-independent if and only if it is a maximum independent set and 1-independent if and only if it is a unique maximum independent set of G. In this paper we mainly investigate connections between perfect independent sets and k-independent as well as super k-independent sets for k = 0 and k = 1.

Topp, J., Graphs with unique minimum edge dominating sets and graphs with unique maximum independent sets of vertices, Discrete Mathematics 12 1 (1993) 199-210. A set I of vertices of a graph G is an independent set if no two vertices of I are adjacent. A set M of edges of G is an edge dominating set if each edge not in M is adjacent to at least one edge in M. We investigate graphs that have unique minimum edge dominating sets. Moreover, we characterize graphs whose total graphs (line graphs) have unique maximum independent sets of vertices.

Suppose that we are given two independent sets I 0 and I r of a graph such that |I 0 | = |I r |, and imagine that a token is placed on each vertex in I 0 . Then, the token jumping problem is to determine whether there exists a sequence of independent sets which transforms I 0 into I r so that each independent set in the sequence results from the previous one by moving exactly one token to another vertex. Therefore, all independent sets in the sequence must be of the same cardinality. This problem is PSPACE-complete even for planar graphs with maximum degree three. In this paper, we first show that the problem is W[1]-hard when parameterized only by the number of tokens. We then give an FPT algorithm for general graphs when parameterized by both the number of tokens and the maximum degree. Our FPT algorithm can be modified so that it finds an actual sequence of independent sets between I 0 and I r with the minimum number of token movements. We finally show that one of the results for token jumping can be extended to a more generalized reconfiguration problem for independent sets, called token addition and removal.

Metastasis via the lymphatics is a major risk factor in squamous cell carcinoma of the oral cavity (OSCC). We sought to determine whether the presence of metastasis in the regional lymph node could be predicted by a gene expression signature of the primary tumor. A total of 18 OSCCs were characterized for gene expression by hybridizing RNA to Affymetrix U133A gene chips. Genes with differential expression were identified using a permutation technique and verified by quantitative RT-PCR and immunohistochemistry. A predictive rule was built using a support vector machine, and the accuracy of the rule was evaluated using crossvalidation on the original data set and prediction of an independent set of four patients. Metastatic primary tumors could be differentiated from nonmetastatic primary tumors by a signature gene set of 116 genes. This signature gene set correctly predicted the four independent patients as well as associating five lymph node metastases from the original patient set with the metastatic primary tumor group. We concluded that lymph node metastasis could be predicted by gene expression profiles of primary oral cavity squamous cell carcinomas. The presence of a gene expression signature for lymph node metastasis indicates that clinical testing to assess risk for lymph node metastasis should be possible.

A graph is CIS if every maximal clique interesects every maximal stable set. Currently, no good characterization or recognition algorithm for the CIS graphs is known. We characterize graphs in which every maximal matching saturates all vertices of degree at least two and use this result to give a structural, efficiently testable characterization of claw-free CIS graphs. We answer in the negative a question of Dobson, Hujdurović, Milanič, and Verret [Vertex-transitive CIS graphs, European J. Combin. 44 (2015) 87-98] asking whether the number of vertices of every CIS graph is bounded from above by the product of its clique and stability numbers. On the positive side, we show that the question of Dobson et al. has an affirmative answer in the case of claw-free graphs.

The accurate orbit prediction of the near-Earth objects is an important requirement for the re-entry and the life time estimation. The method of Kustaanheimo and Stiefel (KS) total energy element equations is one of the powerful methods for orbit prediction. Recently, due to the reentries of large number of risk objects, which posses threat to the human life and property, a great concern is developed in the space scientific community. Consequently, the prediction of risk objects re-entry time and location has got much importance for the proper planning of mitigation strategies and hazard assessment. This paper discusses an integrated procedure for orbit life time prediction combining the KS elements and genetic algorithm (GA). The orbit prediction is carried out by numerically integrating the KS element equations. In this methodology, the ballistic coefficient is estimated from a set of observed orbital parameters in terms of the Two Line Elements (TLE) by minimizing the variance of the predicted re-entry time from different TLE using GA. A software, KSGEN, systematically developed in-house using KS elements and genetic algorithm is utilized for predicting the re-entry time of the risk objects. This software has been effectively used for the prediction of the re-entry time in the past seven re-entry exercise campaigns conducted by the Inter Agency Space Debris Coordination Committee (IADC). The predicted re-entry time matched quite well with the actual re-entry time for all the seven IADC re-entry campaigns. A detailed analysis is carried out with two case studies.

In this paper, we deal with the problem of finding quasi-independent sets in graphs. This problem is formally defined in three versions, which are shown to be polynomially equivalent. The one that looks most general, namely, f -QIS, consists of, given a graph and a non-decreasing function f , finding a maximum size subset Q of the vertices of the graph, such that the number of edges in the induced subgraph is less than or equal to f (|Q|). For this problem, we show an exact solution method that runs within time O * (2 d-27/23 d+1 n ) on graphs of average degree bounded by d. For the most specifically defined γ-QIS and k-QIS problems, several results on complexity and approximation are shown, and greedy algorithms are proposed, analyzed and tested.

In this paper, we deal with the problem of finding quasi-independent sets in graphs. This problem is formally defined in three versions, which are shown to be polynomially equivalent. The one that looks most general, namely, f -QIS, consists of, given a graph and a non-decreasing function f , finding a maximum size subset Q of the vertices of the graph, such that the number of edges in the induced subgraph is less than or equal to f (|Q|). For this problem, we show an exact solution method that runs within time O * (2 d-27/23 d+1 n ) on graphs of average degree bounded by d. For the most specifically defined γ-QIS and k-QIS problems, several results on complexity and approximation are shown, and greedy algorithms are proposed, analyzed and tested.

Let A be an m × n matrix in which the entries of each row are all distinct. Drisko [4] showed that, if m ≥ 2n -1, then A has a transversal : a set of n distinct entries with no two in the same row or column. We generalize this to matrices with entries in a matroid. For such a matrix A, we show that if each row of A forms an independent set, then we can require the transversal to be independent as well. We determine the complexity of an algorithm based on the proof of this result. Lastly, we observe that m ≥ 2n -1 appears to force the existence of not merely one but many transversals. We discuss a number of conjectures related to this observation (some of which involve matroids and some of which do not).

Suppose that we are given two independent sets I b and Ir of a graph such that |I b | = |Ir|, and imagine that a token is placed on each vertex in I b . Then, the sliding token problem is to determine whether there exists a sequence of independent sets which transforms I b into Ir so that each independent set in the sequence results from the previous one by sliding exactly one token along an edge in the graph. This problem is known to be PSPACE-complete even for planar graphs, and also for bounded treewidth graphs. In this paper, we thus study the problem restricted to trees, and give the following three results: (1) the decision problem is solvable in linear time; (2) for a yes-instance, we can find in quadratic time an actual sequence of independent sets between I b and Ir whose length (i.e., the number of token-slides) is quadratic; and (3) there exists an infinite family of instances on paths for which any sequence requires quadratic length.

The administration of dextran-conjugated cytosine arabinonucleoside (araC) to BALB/c mice at various times prior to but not subsequent to immunization with native dextran renders mice unresponsive to this thymic-independent antigen. These results demonstrate that the primary immune response to an antigen can be selectively and efficiently suppressed or eliminated in vivo by the delivery of a single dose of an appropriate antigen-cytotoxic drug conjugate. Evidence presented here indicates that the dextran-araC conjugate (toxogen) acts directly and selectively upon unprimed dextran-specific antibody-forming cell precursors, presumably by binding to their receptors and subsequent internalization of the resultant receptor-toxogen complexes. The resistance of antigen-primed mice to the cytotoxic effect of the toxogen could result from the failure of dextran-primed cells to reexpress antigen-specific receptors, from an alternative processing of the toxogen, or from the inability of the an...

A graph is CIS if every maximal clique interesects every maximal stable set. Currently, no good characterization or recognition algorithm for the CIS graphs is known. We characterize graphs in which every maximal matching saturates all vertices of degree at least two and use this result to give a structural, efficiently testable characterization of claw-free CIS graphs. We answer in the negative a question of Dobson, Hujdurović, Milanič, and Verret [Vertex-transitive CIS graphs, European J. Combin. 44 (2015) 87-98] asking whether the number of vertices of every CIS graph is bounded from above by the product of its clique and stability numbers. On the positive side, we show that the question of Dobson et al. has an affirmative answer in the case of claw-free graphs.

Let S be a set of linear inequalities that determine a bounded polyhedron P. The closure of S is the smallest set of inequalities that contains S and is closed under two operations: (i) taking linear combinations of inequalities, (ii) replacing an inequality C uixi 5 a~, wRere uI, a2, . . . . an are integers, by the inequality Cajxj <_ a with o 2 [a~]. Obviously, if integersxl, x2, . . . . xn satisfy all the inequalities in S, then they satisfy also all inequalities in the closure of S. Conversely, let Zcixi < CO hold for al'1 choices of integers x1, x2, . . . . X8$ that satisfy all the inequalitities in S. Then we prove that 'ccixi _ < CO belongs to the closure of S. To each integer linear programming problem, we assign a nonnegative integer, called its rank. (The rank is the minimum number of iterations of the operation (ii) that are required in order to eliminate the integrality constraint.) We prove that there is no upper bound on the rank of problems arising from the search for largest independent sets in graphs.

Multiple-interval graphs are a natural generalization of interval graphs where each vertex may have more than one interval associated with it. Many applications of interval graphs also generalize to multiple-interval graphs, often allowing for more robustness in the modeling of the specific application. With this motivation in mind, a recent systematic study of optimization problems in multiple-interval graphs was initiated. In this sequel, we study multiple-interval graph problems from the perspective of parameterized complexity. The problems under consideration are k-Independent Set, k-Dominating Set, and k-Clique, which are all known to be W[1]-hard for general graphs, and NP-complete for multiple-interval graphs. We prove that k-Clique is in FPT, while k-Independent Set and k-Dominating Set are both W[1]-hard. We also prove that k-Independent Dominating Set, a hybrid of the two above problems, is also W[1]-hard. Our hardness results hold even when each vertex is associated with at most two intervals, and all intervals have unit length. Furthermore, as an interesting byproduct of our hardness results, we develop a useful technique for showing W[1]-hardness via a reduction from the k-Multicolored Clique problem, a variant of k-Clique. We believe this technique has interest in its own right, as it should help in simplifying W[1]-hardness results which are notoriously hard to construct and technically tedious.

In this paper we consider the problem of finding a graph separator of a given size that decomposes the graph into the maximum number of connected components. We present the picture of the computational complexity and the approximability of this problem for several natural classes of graphs. We first provide an overview of the hardness of approximation of this problem, which stems mainly from its close relation to the INDEPENDENT SET and to the MAXIMUM CLIQUE problem. Next, we show that the problem is solvable in polynomial time for interval graphs and graphs of bounded treewidth. We also show that MAXINUM COMPONENTS is fixed-parameter tractable on planar graphs with the size of the separator as the parameter. Our main contribution is the derivation of an efficient polynomial-time approximation scheme for the problem on planar graphs.

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We derive bounds on the size of an independent set based on eigenvalues. This generalizes a result due to Delsarte and Hoffman. We use this to obtain new bounds on the independence number of the Erdős-Rényi graphs. We investigate further properties of our bounds, and show how our results on the Erdős-Rényi graphs can be extended to other polarity graphs.

As a noninvasive colorectal cancer (CRC) screening test, a multi-marker first generation stool DNA (sDNA V 1.0) test is superior to guaiac-based fecal occult blood tests. An improved sDNA assay (version 2), utilizing only two markers, hypermethylated vimentin gene (hV) and a two site DNA integrity assay (DY), demonstrated in a training set (phase 1a) an even higher sensitivity (88%) for CRC with a specificity of 82%. To validate in an independent set of patients (phase 1b) the sensitivity and specificity of sDNA version 2 for CRC. Forty-two patients with CRC and 241 subjects with normal colonoscopy (NC) provided stool samples, to which they immediately added DNA stabilizing buffer, and mailed their specimen to the laboratory. DNA was purified using gel-based capture, and analyzed for hV and DY using methods identical to those previously published. Using the same cutpoints as the 1a training set (N = 162; 40 CRCs, 122 normals), hV demonstrated a higher and DY a slightly lower sensiti...

Let G m;n be the m n grid graph. That is, the vertices of G m;n are the (m+1)(n+1) points (i; j) (0 i m; 0 j n) in the plane, and its edges are all of the pairs (i; j); (i 0 ; j 0 ) of vertices for which ji ? i 0 j + jj ? j 0 j = 1. Let f(m; n) be the number of independent sets of vertices in G m;n . We study the growth of f(m; n). The gure below shows an independent set S in G 4;6 .

Preprocessing of near-infrared spectra to remove unwanted, i.e., non-related spectral variation and selection of informative wavelengths is considered to be a crucial step prior to the construction of a quantitative calibration model. The standard methodology when comparing various preprocessing techniques and selecting different wavelengths is to compare prediction statistics computed with an independent set of data not used to make the actual calibration model. When the errors of reference value are large, no such values are available at all, or only a limited number of samples are available, other methods exist to evaluate the preprocessing method and wavelength selection. In this work we present a new indicator (SE) that only requires blank sample spectra, i.e., spectra of samples that are mixtures of the interfering constituents (everything except the analyte), a pure analyte spectrum, or alternatively, a sample spectrum where the analyte is present. The indicator is based on c...

Correctly determining a company's market worth during an entire year or a certain period presents a difficulty to decision-makers. In the case of the merger of companies, the need performs heavier when both the companies' owners are attracted to establishing a fair price at the optimal time to merge. The effectiveness of representing, connecting and manipulating both uncertainty and periodicity information becomes highly required. Hence, study and nhance some properties and conditions of the algebraic structure of complex hesitant fuzzy graphs. Therefore, the degree of composition between two complex hesitant fuzzy graphs is proposed. Also, the formal definitions of union, joint and complement are presented to be covered in the realm of complex hesitant fuzzy graphs. A real-life application is illustrated to show the relation between vertices and edges in the form of complex hesitant fuzzy graphs.

For a graph, the SK index is equal to the half of the sum of the degrees of the vertices, the SK1 index is equal to the half of the product of the degrees of the vertices, and the SK2 index is equal to the half of the square of the sum of the degrees of the vertices. This paper shows a simple and unified approach to the greatest SK indices for unicyclic graphs by using some transformations and characterizes these graphs with the first, second, and third SK indices having order r ≥ 5 and girth g ≥ 3, where girth is the length of the shortest cycle in a graph.

PDCD1, an immunoreceptor involved in peripheral tolerance has previously been shown to be genetically associated with systemic lupus erythematosus (SLE). PDCD1 has two ligands whose genes are located in close proximity on chromosome 9p24. Our attention was drawn to these ligands after finding suggestive linkage to a marker (gata62f03, Z ¼ 2.27) located close to their genes in a genome scan of Icelandic families multiplex for SLE. Here, we analyse Swedish trios (N ¼ 149) for 23 single nucleotide polymorphisms (SNPs) within the genes of the PDCD1 ligands. Initially, indication of association to eight SNPs was observed, and these SNPs were therefore also analysed in Mexican trios (N ¼ 90), as well as independent sets of patients and controls from Sweden (152 patients, 448 controls) and Argentina (288 patients, 288 controls). We do not find support for genetic association to SLE. This is the first genetic study of SLE and the PDCD1 ligands and the lack of association in several cohorts implies that these genes are not major risk factors for SLE.

We present a family of number sequences which interpolates between the sequences Bn; of Bell numbers, and n!. It is deÿned in terms of permutations with forbidden patterns or subsequences. The introduction, as a parameter, of the number m of right-to-left minima yields an interpolation between Stirling numbers of the second kind S(n; m) and of the ÿrst kind (signless) c(n; m). Moreover, q-counting the restricted permutations by special inversions gives an interpolation between variants of the usual q-analogues of these numbers. Nous prà esentons une famille de suites de nombres qui interpole entre la suite Bn des nombres de Bell et la suite n!. Cette famille est dà eÿnie en termes de permutations  a motifs interdits. L'introduction comme param etre du nombre d'à elà ements saillants minimums de gauche  a droite donne une interpolation plus ÿne entre les nombres de Stirling de deuxi eme esp ece S(n; m) et de premi ere esp ece (sans signe) c(n; m). De plus, un q-comptage de ces permutations selon des inversions particuli eres donne une interpolation entre des variantes des q-analogues habituels de ces nombres.

The transformation of normal cells into cancerous cells is an evolutionary process. Populations of precancerous cells reproduce, mutate, and compete for resources. Some of these mutations eventually lead to cancer. We calculate the probability of developing cancer under a set of simplifying assumptions and then elaborate these calculations, culminating in a simple simulation of the cell dynamics. The agent-based model allows us to examine the interactions of mutations critical for the development of cancer that are either evolutionarily neutral or selective. We can also examine the interaction of these mutations with a “mutator phenotype” derived from mutations that raise the mutation rate for the entire cell. The simulations suggest that there must be at least two selectively neutral mutations necessary for the development of cancer and that preventive treatments will be most effective when they increase this number. The model also suggests that selective mutations facilitate the d...

The calibration and application of a simulation-based dynamic traffic assignment (DTA) model on a portion of the city of Calgary road network in Alberta, Canada, are discussed. The DTA model iteratively reassigns flow to paths by using the method of successive averages on the basis of travel times obtained with a traffic simulation model. The original subnetwork extracted from a regional planning model was enriched by a great increase in the number of zones. The DTA origin-destination matrix was estimated from an extensive database of turning movement counts via a trip generation/distribution model and a matrix adjustment algorithm. The network topology was enhanced by the addition of an interchange and a more precise representation of arterial intersections, including traffic signal control plans. A set of 1-h turning counts was used to calibrate the DTA model by adjusting local parameters such as gap-acceptance values, as well as global parameters such as average vehicle length. T...

Single nucleotide polymorphisms (SNPs) are the most frequent form of human genetic variation. They are of fundamental importance for a variety of applications including medical diagnostic and drug design. They also provide the highest-resolution genomic fingerprint for tracking disease genes. This paper is devoted to algorithmic problems related to computational SNPs validation based on genome assembly of diploid organisms. In diploid genomes, there are two copies of each chromosome. A description of the SNPs sequence information from one of the two chromosomes is called SNPs haplotype. The basic problem addressed here is the Haplotyping, i.e., given a set of SNPs prospects inferred from the assembly alignment of a genomic region of a chromosome, find the maximally consistent pair of SNPs haplotypes by removing data "errors" related to DNA sequencing errors, repeats, and paralogous recruitment. In this paper, we introduce several versions of the problem from a computational point of view. We show that the general SNPs Haplotyping Problem is NP-hard for mate-pairs assembly data, and design polynomial time algorithms for fragment assembly data. We give a network-flow based polynomial algorithm for the Minimum Fragment Removal Problem, and we show that the Minimum SNPs Removal problem amounts to finding the largest independent set in a weakly triangulated graph.

Probe-based confocal laser endomicroscopy (pCLE) is an imaging technique that allows real-time in vivo histological assessment of Barrett&#39;s esophagus (BE). The objectives of this study were to create and test novel pCLE criteria for dysplastic BE (phase I), and to evaluate accuracy, interobserver variability, and learning curve in dysplasia prediction (phase II) using these criteria. In phase I, using 50 pCLE videos, a pCLE expert and gastrointestinal pathologist formulated new BE criteria by consensus. These criteria were tested and refined in an independent set of 30 pCLE videos. In phase II, a formal training session for all assessors (three each experts/trainees) was conducted. Finally, using 75 testing videos, each video was interpreted as dysplasia (high-grade dysplasia (HGD)/cancer) vs. no dysplasia and the assessors&#39; confidence in interpretation was noted. Interobserver agreement and accuracy (95% confidence interval (CI)) were determined for BE histology prediction....

A stable set of a graph is a set of mutually non-adjacent vertices. The determination of a maximum size stable set, which is called maximum stable set, and the determination of its size, which is called stability number, are central combinatorial optimization problems. However, given a nonnegative integer k, to determine if a graph G has a stable set of size k is NP-complete. In this paper we deal with graphs for which the stability number can be determined by solving a convex quadratic programming problem. Such graphs were introduced in [13] and are called graphs with convex-gP stability number. A few algorithmic techniques for the recognition of this type of graphs in particular families are presented.

We study the maximum weight independent sets of links between nodes distributed as a spatial Poisson process in an infinite plane. Three different definitions of the weight of a link are considered, leading to slight variations of what is essentially a spatial reuse problem in wireless multihop networks. A simple Boolean interference model is assumed with the interference radius equaling the transmission radius. We study both the case where the transmission radius is fixed and the case where it can be reduced (by power control) so as to just reach the destination to minimize the interference. For the case of a fixed transmission radius, we give asymptotic results for the low density regime and present a rudimentary analysis for the high density asymptotics. The main contribution of this paper is in the numerical results for the maximum weight for the considered infinite networks and in thus establishing some previously unknown parameters of stochastic geometry. For instance, we find that in the unweighted case, just counting the number of independent links, the maximum possible packing is 0.322 links per node attained with the mean neighborhood size of 2.73. The results are obtained by the Moving Window Algorithm that is able to find the maximum weight independent set in a strip of limited height but unlimited length. By studying the results as the function of the height of the strip, we are able to extrapolate to the infinite plane.

We study the maximum weight independent sets of links between nodes distributed as a spatial Poisson process in an infinite plane. Three different definitions of the weight of a link are considered, leading to slight variations of what is essentially a spatial reuse problem in wireless multihop networks. A simple Boolean interference model is assumed with the interference radius equaling the transmission radius. We study both the case where the transmission radius is fixed and the case where it can be reduced (by power control) so as to just reach the destination to minimize the interference. For the case of a fixed transmission radius, we give asymptotic results for the low density regime and present a rudimentary analysis for the high density asymptotics. The main contribution of this paper is in the numerical results for the maximum weight for the considered infinite networks and in thus establishing some previously unknown parameters of stochastic geometry. For instance, we find that in the unweighted case, just counting the number of independent links, the maximum possible packing is 0.322 links per node attained with the mean neighborhood size of 2.73. The results are obtained by the Moving Window Algorithm that is able to find the maximum weight independent set in a strip of limited height but unlimited length. By studying the results as the function of the height of the strip, we are able to extrapolate to the infinite plane.

We report measurements of Ω M , Ω Λ , and w from eleven supernovae at z = 0.36-0.86 with high-quality lightcurves measured using WFPC2 on the HST. This is an independent set of highredshift supernovae that confirms previous supernova evidence for an accelerating Universe. The high-quality lightcurves available from photometry on WFPC2 make it possible for these eleven supernovae alone to provide measurements of the cosmological parameters comparable in statistical weight to the previous results. Combined with earlier Supernova Cosmology Project data, the new supernovae yield a measurement of the mass density Ω M = 0.25 +0.07 -0.06 (statistical) ±0.04 (identified systematics), or equivalently, a cosmological constant of Ω Λ = 0.75 +0.06 -0.07 (statistical) ±0.04 (identified systematics), under the assumptions of a flat universe and that the dark energy equation of state parameter has a constant value w = -1. When the supernova results are combined with independent flat-universe measurements of Ω M from CMB and galaxy redshift distortion data, they provide a measurement of w = -1.05 +0.15 -0.20 (statistical) ±0.09 (identified systematic), if w is assumed to be constant in time. In addition to high-precision lightcurve measurements, the new data offer greatly improved color measurements of the high-redshift supernovae, and hence improved host-galaxy extinction estimates. These extinction measurements show no anomalous negative E(B-V ) at high redshift. The precision of the measurements is such that it is possible to perform a host-galaxy extinction correction directly for individual supernovae without any assumptions or priors on the parent E(B-V ) distribution. Our cosmological fits using full extinction corrections confirm that dark energy is required with P (Ω Λ > 0) > 0.99, a result consistent with previous and current supernova analyses which rely upon the identification of a low-extinction subset or prior assumptions concerning the intrinsic extinction distribution. Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W.M. Keck Foundation. Based in part on observations obtained at the WIYN Observatory, which is a joint facility of the

Background: Prediction of protein subcellular localization generally involves many complex factors, and using only one or two aspects of data information may not tell the true story. For this reason, some recent predictive models are deliberately designed to integrate multiple heterogeneous data sources for exploiting multi-aspect protein feature information. Gene ontology, hereinafter referred to as GO, uses a controlled vocabulary to depict biological molecules or gene products in terms of biological process, molecular function and cellular component. With the rapid expansion of annotated protein sequences, gene ontology has become a general protein feature that can be used to construct predictive models in computational biology. Existing models generally either concatenated the GO terms into a flat binary vector or applied majority-vote based ensemble learning for protein subcellular localization, both of which can not estimate the individual discriminative abilities of the three aspects of gene ontology. Results: In this paper, we propose a Gene Ontology Based Transfer Learning Model (GO-TLM) for large-scale protein subcellular localization. The model transfers the signature-based homologous GO terms to the target proteins, and further constructs a reliable learning system to reduce the adverse affect of the potential false GO terms that are resulted from evolutionary divergence. We derive three GO kernels from the three aspects of gene ontology to measure the GO similarity of two proteins, and derive two other spectrum kernels to measure the similarity of two protein sequences. We use simple non-parametric cross validation to explicitly weigh the discriminative abilities of the five kernels, such that the time & space computational complexities are greatly reduced when compared to the complicated semi-definite programming and semi-indefinite linear programming. The five kernels are then linearly merged into one single kernel for protein subcellular localization. We evaluate GO-TLM performance against three baseline models: MultiLoc, MultiLoc-GO and Euk-mPLoc on the benchmark datasets the baseline models adopted. 5-fold cross validation experiments show that GO-TLM achieves substantial accuracy improvement against the baseline models: 80.38% against model Euk-mPLoc 67.40% with 12.98% substantial increase; 96.65% and 96.27% against model MultiLoc-GO 89.60% and 89.60%, with 7.05% and 6.67% accuracy increase on dataset MultiLoc plant and dataset MultiLoc animal, respectively; 97.14%, 95.90% and 96.85% against model MultiLoc-GO 83.70%, 90.10% and 85.70%, with accuracy increase 13.44%, 5.8% and 11.15% on dataset BaCelLoc plant, dataset BaCelLoc fungi and dataset BaCelLoc animal respectively. For BaCelLoc independent sets, GO-TLM achieves 81.25%, 80.45% and 79.46% on dataset BaCelLoc plant holdout, dataset BaCelLoc plant holdout and dataset BaCelLoc animal holdout, respectively, as compared against baseline model MultiLoc-GO 76%, 60.00% and 73.00%, with accuracy increase 5.25%, 20.45% and 6.46%, respectively. Conclusions: Since direct homology-based GO term transfer may be prone to introducing noise and outliers to the target protein, we design an explicitly weighted kernel learning system (called Gene Ontology Based Transfer Learning Model, GO-TLM) to transfer to the target protein the known knowledge about related homologous proteins, which can reduce the risk of outliers and share knowledge between homologous proteins, and thus achieve better predictive performance for protein subcellular localization. Cross validation and independent test experimental results show that the homology-based GO term transfer and explicitly weighing the GO kernels substantially improve the prediction performance.

Background: Prediction of protein subcellular localization generally involves many complex factors, and using only one or two aspects of data information may not tell the true story. For this reason, some recent predictive models are deliberately designed to integrate multiple heterogeneous data sources for exploiting multi-aspect protein feature information. Gene ontology, hereinafter referred to as GO, uses a controlled vocabulary to depict biological molecules or gene products in terms of biological process, molecular function and cellular component. With the rapid expansion of annotated protein sequences, gene ontology has become a general protein feature that can be used to construct predictive models in computational biology. Existing models generally either concatenated the GO terms into a flat binary vector or applied majority-vote based ensemble learning for protein subcellular localization, both of which can not estimate the individual discriminative abilities of the three aspects of gene ontology. Results: In this paper, we propose a Gene Ontology Based Transfer Learning Model (GO-TLM) for large-scale protein subcellular localization. The model transfers the signature-based homologous GO terms to the target proteins, and further constructs a reliable learning system to reduce the adverse affect of the potential false GO terms that are resulted from evolutionary divergence. We derive three GO kernels from the three aspects of gene ontology to measure the GO similarity of two proteins, and derive two other spectrum kernels to measure the similarity of two protein sequences. We use simple non-parametric cross validation to explicitly weigh the discriminative abilities of the five kernels, such that the time & space computational complexities are greatly reduced when compared to the complicated semi-definite programming and semi-indefinite linear programming. The five kernels are then linearly merged into one single kernel for protein subcellular localization. We evaluate GO-TLM performance against three baseline models: MultiLoc, MultiLoc-GO and Euk-mPLoc on the benchmark datasets the baseline models adopted. 5-fold cross validation experiments show that GO-TLM achieves substantial accuracy improvement against the baseline models: 80.38% against model Euk-mPLoc 67.40% with 12.98% substantial increase; 96.65% and 96.27% against model MultiLoc-GO 89.60% and 89.60%, with 7.05% and 6.67% accuracy increase on dataset MultiLoc plant and dataset MultiLoc animal, respectively; 97.14%, 95.90% and 96.85% against model MultiLoc-GO 83.70%, 90.10% and 85.70%, with accuracy increase 13.44%, 5.8% and 11.15% on dataset BaCelLoc plant, dataset BaCelLoc fungi and dataset BaCelLoc animal respectively. For BaCelLoc independent sets, GO-TLM achieves 81.25%, 80.45% and 79.46% on dataset BaCelLoc plant holdout, dataset BaCelLoc plant holdout and dataset BaCelLoc animal holdout, respectively, as compared against baseline model MultiLoc-GO 76%, 60.00% and 73.00%, with accuracy increase 5.25%, 20.45% and 6.46%, respectively. Conclusions: Since direct homology-based GO term transfer may be prone to introducing noise and outliers to the target protein, we design an explicitly weighted kernel learning system (called Gene Ontology Based Transfer Learning Model, GO-TLM) to transfer to the target protein the known knowledge about related homologous proteins, which can reduce the risk of outliers and share knowledge between homologous proteins, and thus achieve better predictive performance for protein subcellular localization. Cross validation and independent test experimental results show that the homology-based GO term transfer and explicitly weighing the GO kernels substantially improve the prediction performance.

We investigate the minimum independent dominating set in perturbed graphs g(G, p) of input graph G = (V, E), obtained by negating the existence of edges independently with a probability p > 0. The minimum independent dominating set (MIDS) problem does not admit a polynomial running time approximation algorithm with worst-case performance ratio of n 1- for any > 0. We prove that the size of the minimum independent dominating set in g(G, p), denoted as i(g(G, p)), is asymptotically almost surely in Θ(log |V |). Furthermore, we show that the probability of i(g(G, p)) ≥ 4|V | p is no more than 2 -|V | , and present a simple greedy algorithm of proven worst-case performance ratio 4|V | p and with polynomial expected running time.

This paper studies k-claw-free graphs, exploring the connection between an extremal combinatorics question and the power of a convex program in approximating the maximum-weight independent set in this graph class. For the extremal question, we consider the notion, that we call conditional χ-boundedness of a graph: Given a graph G that is assumed to contain an independent set of a certain (constant) size, we are interested in upper bounding the chromatic number in terms of the clique number of G. This question, besides being interesting on its own, has algorithmic implications (which have been relatively neglected in the literature) on the performance of SDP relaxations in estimating the value of maximum-weight independent set. For k = 3, Chudnovsky and Seymour (JCTB 2010) prove that any 3-clawfree graph G with an independent set of size three must satisfy χ(G) ≤ 2ω(G). Their result implies a factor 2-estimation algorithm for the maximum weight independent set via an SDP relaxation (providing the first non-trivial result for maximum-weight independent set in such graphs via a convex relaxation). An obvious open question is whether a similar conditional χ-boundedness phenomenon holds for any k-claw-free graph. Our main result answers this question negatively. We further present some evidence that our construction could be useful in studying more broadly the power of convex relaxations in the context of approximating maximum weight independent set in k-claw free graphs. In particular, we prove a lower bound on families of convex programs that are stronger than known convex relaxations used algorithmically in this context.

A set S ⊆ V (G) is an independent transversal dominating set of a graph G if S is a dominating set of G and intersects every maximum independent set of G. An independent transversal dominating set which is a total dominating set is an independent transversal total dominating set. The minimum cardinality γit(G) (resp. γitt(G)) of an independent transversal dominating set (resp. independent transversal total dominating set) of G is the independent transversal domination number (resp. independent transversal total domination number) of G. In this paper, we show that for every positive integers a and b with 5 ≤ a ≤ b ≤ 2a − 2, there exists a connected graph G for which γit(G) = a and γitt(G) = b. We also study these two concepts in graphs which are the join, corona or composition of graphs.

We introduce the closed domination in graphs. Some interesting relationships are known between domination and closed domination and between closed domination and the independent domination. It is also shown that any triple m,k and n of positive integers with 3≤m≤k≤n are realizable as the domination number, closed domination number and independent domination number, respectively, of a graph. In this paper, we also investigate the closed domination in the corona and composition of graphs.

We pose robust matching with parametric and non-parametric constraints as the problem of finding a stable independent set (SIS) in an oriented graph whose vertices are all possible correspondences, whose edges capture the structure of the constraints and whose edge orientation represents pairwise comparison 'is better' based on correspondence quality, including the uncertainty of this comparison. We show SIS possess properties of both robustness and weak optimality. The main contribution of this paper is algorithmic speedup that results from exploiting the dependence between the standard uniqueness constraint and the parametric constraint. The general theory is demonstrated on the example of image stitching using homography model. The algorithm needs at most k N 2 calls of a procedure testing if two ellipse correspondences are consistent with a general homography. The previous known SIS algorithm needed O(N 4 ) tests. Experiments show the method gives good results and is fast in practice with k ≈ 0.3.

Efficient multicast congestion control (MCC) is one of the critical components required to enable the IP multicast deployment over the Internet. Previously proposed MCC schemes can be categorized in two: single-rate or multi-rate. Single-rate schemes make all recipients get data at a common rate allowed by the slowest receiver, but are relatively simple. Multi-rate schemes allow of heterogeneous receive rates and thus provide better scalability, but rely heavily on frequent updates to group membership state in the routers. A recent work by Kwon and Byers, combined these two methods and provided a multi-rate scheme by means of single-rate schemes with relatively low complexity. In this paper, we propose a new scheme called generalized multicast congestion control (GMCC). GMCC provides multi-rate features at low complexity by using a set of independent single-rate sub-sessions (a.k.a layers) as building blocks. The scheme is named GMCC because single-rate MCC is just one of its special cases. Unlike the earlier work by Kwon and Byers, GMCC does not have the drawback of static configuration of the source which may not match with the dynamic network situations. GMCC is fully adaptive in that (i) it does not statically set a particular range for the sending rates of layers, and (ii) it eliminates redundant layers when they are not needed. Receivers can subscribe to different subsets of the available layers and hence can always obtain different throughput. While no redundant layers are used, GMCC allows receivers to activate a new layer in case existing layers do not accommodate the needs of the actual receivers.

In this work we deal with sandwich graphs G = (V, E) and present the notion of vertices f-controlled by a subset M ⊆ V. We introduce the generalized maxcontrolled set problem (gmcsp), where gaps and positive weights are associated to each vertex of V. In this case, the objective is find a sandwich graph G in order to maximize the sum of the weights associated to all vertices f-controlled by M. We present a 1 2-approximation algorithm for the gmcsp and a new procedure for finding feasible solutions based on a linear relaxation. The best solution is then used as starting point in a local search procedure (Tabu Search with Path Relinking). Finally, we present some computational results and compare the performance of our heuristics with the optimum solution value of some instances of the problem.

Staton proved that every triangle-free graph on n vertices with maximum degree three has an independent set of size at least 5n/14. A simpler proof was found by Jones. We give a yet simpler proof, and use it to design a linear-time algorithm to find such an independent set. Let G be a triangle-free graph on n vertices with maximum degree three. By Brooks' theorem [1] G is 3-colorable; considering the largest color class, it follows that G has an independent set of size at least n/3. The coloring result is clearly best possible, but can we do better in terms of independent sets? Staton [6] proved that, in fact, the bound can be improved to 5n/14, and this is best possible because, as noticed by Fajtlowicz [2], the generalized Petersen graph P (7, 2) has fourteen vertices and no independent set of size six. Jones [5] found a simpler proof of Staton's result. Griggs and Murphy [3] designed a linear-time algorithm to find an independent set in G of size at least 5(n − k)/14, where k is the number of components of G that are 3-regular. The objective of this paper is to give a yet simpler proof of Staton's result, and to design a linear-time algorithm to find an independent set in G of size at least 5n/14. Graphs are finite and simple (that is, they have no loops or parallel edges). A block is either a 2-connected graph, or a complete graph on at most two vertices. A block of a graph G is a maximal subgraph of G that is a block. A pentagon is a cycle of length five. (Paths and cycles have no "repeated" vertices.) A block B is said to be difficult if it is isomorphic to a pentagon or L, where L is the graph obtained by subdividing both edges of a perfect matching of K 4 twice; it has eight vertices, ten edges and independent sets of size three. A graph G is said to be difficult if every block of G is either difficult or is an edge between two difficult blocks. We define λ(G) to be the number of components of G that are difficult. Our main result reads as follows. Theorem. Every triangle-free graph G with maximum degree at most three has an independent set of size at least 1 7 (4 |V (G)| − |E(G)| − λ(G)). Since every difficult component has at least two vertices of degree two, we deduce Staton's theorem. Corollary. Every triangle-free graph on n vertices with maximum degree at most three has an independent set of size at least 5n/14. We offer the following conjecture, which would also imply the corollary. The fractional chromatic number of a graph G is the infimum of all a/b such that to every vertex of G one can assign a subset of {1, 2,. .. , a} of size b in such a way that adjacent vertices are assigned disjoint sets.

Staton proved that every triangle-free graph on n vertices with maximum degree three has an independent set of size at least 5n/14. A simpler proof was found by Jones. We give a yet simpler proof, and use it to design a linear-time algorithm to find such an independent set. Let G be a triangle-free graph on n vertices with maximum degree three. By Brooks' theorem [1] G is 3-colorable; considering the largest color class, it follows that G has an independent set of size at least n/3. The coloring result is clearly best possible, but can we do better in terms of independent sets? Staton [6] proved that, in fact, the bound can be improved to 5n/14, and this is best possible because, as noticed by Fajtlowicz [2], the generalized Petersen graph P (7, 2) has fourteen vertices and no independent set of size six. Jones [5] found a simpler proof of Staton's result. Griggs and Murphy [3] designed a linear-time algorithm to find an independent set in G of size at least 5(n − k)/14, where k is the number of components of G that are 3-regular. The objective of this paper is to give a yet simpler proof of Staton's result, and to design a linear-time algorithm to find an independent set in G of size at least 5n/14. Graphs are finite and simple (that is, they have no loops or parallel edges). A block is either a 2-connected graph, or a complete graph on at most two vertices. A block of a graph G is a maximal subgraph of G that is a block. A pentagon is a cycle of length five. (Paths and cycles have no "repeated" vertices.) A block B is said to be difficult if it is isomorphic to a pentagon or L, where L is the graph obtained by subdividing both edges of a perfect matching of K 4 twice; it has eight vertices, ten edges and independent sets of size three. A graph G is said to be difficult if every block of G is either difficult or is an edge between two difficult blocks. We define λ(G) to be the number of components of G that are difficult. Our main result reads as follows. Theorem. Every triangle-free graph G with maximum degree at most three has an independent set of size at least 1 7 (4 |V (G)| − |E(G)| − λ(G)). Since every difficult component has at least two vertices of degree two, we deduce Staton's theorem. Corollary. Every triangle-free graph on n vertices with maximum degree at most three has an independent set of size at least 5n/14. We offer the following conjecture, which would also imply the corollary. The fractional chromatic number of a graph G is the infimum of all a/b such that to every vertex of G one can assign a subset of {1, 2,. .. , a} of size b in such a way that adjacent vertices are assigned disjoint sets.

Let G = (V, E) be a simple graph. A set S ⊆ V is independent set of G, if no two vertices of S are adjacent. The independence number α(G) is the size of a maximum independent set in the graph. Let R be a commutative ring with nonzero identity and I an ideal of R. The zero-divisor graph of R, denoted by Γ(R), is an undirected graph whose vertices are the nonzero zero-divisors of R and two distinct vertices x and y are adjacent if and only if xy = 0. Also the ideal-based zero-divisor graph of R, denoted by Γ I (R), is the graph which vertices are the set {x ∈ R\I|xy ∈ I for some y ∈ R\I} and two distinct vertices x and y are adjacent if and only if xy ∈ I. In this paper we study the independent sets and the independence number of Γ(R) and Γ I (R).

The independence polynomial of a graph is the generating polynomial for the number of independent sets of each size. Two graphs are said to be independence equivalent if they have equivalent independence polynomials. We extend previous work by showing that independence equivalence class of every odd path has size 1, while the class can contain arbitrarily many graphs for even paths. We also prove that the independence equivalence class of every even cycle consists of two graphs when n ≥ 2 except the independence equivalence class of C6 which consists of three graphs. The odd case remains open, although, using irreducibility results from algebra, we were able show that for a prime p ≥ 5 and n ≥ 1 the independence equivalence class of Cpn consists of only two graphs.

A graph is well-covered if every independent set can be extended to a maximum independent set. We show that it is co-NP-complete to determine whether an arbitrary graph is well-covered, even when restricted to the family of circulant graphs. Despite the intractability of characterizing the complete set of well-covered circulant graphs, we apply the theory of independence polynomials to show that several families of circulants are indeed well-covered. Since the lexicographic product of two well-covered circulants is also a well-covered circulant, our partial characterization theorems enable us to generate infinitely many families of well-covered circulants previously unknown in the literature.

The purpose of the independent set interdiction problem in the weighted graph G is to determine a set of vertices R * such that the weight of the maximum independent set in G − R * is minimized. We define an approximate solution for this problem. Then, an upper bound for the relative error of this problem is obtained. We show that the limit of the relative error of the independent set interdiction problem in some subclasses of the generalized Petersen graphs is zero as the number of vertices tends to infinity.