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For a simple, undirected, connected graph G, a function h:V→{0,1,2} is called a total Roman {2}-dominating function (TR2DF) if for every vertex v in V with weight 0, either there exists a vertex u in NG(v) with weight 2, or at least two vertices x,y in NG(v) each with weight 1, and the subgraph induced by the vertices with weight more than zero has no isolated vertices. The weight of TR2DF h is ∑p∈Vh(p). The problem of determining TR2DF of minimum weight is called minimum total Roman \{2\}-domination problem (MTR2DP). We show that MTR2DP is polynomial time solvable for bounded treewidth graphs, threshold graphs and chain graphs. We design a 2(ln(Δ−0.5)+1.5)-approximation algorithm for the MTR2DP and show that the same cannot have (1−δ)ln|V| ratio approximation algorithm for any δ>0 unless P=NP. Next, we show that MTR2DP is APX-hard for graphs with Δ=4. Finally, we show that the domination and TR2DF problems are not equivalent in computational complexity aspects. [ABSTRACT FROM AUTHOR]

Suppose G=(V,E) is a graph with the vertex set V(G) and edge set E(G) we defined a labeling f:V∪E→{1,2,...,k} to be an edge irregular total k-labeling of graph G if for every two different edge e and f there is wt(e)≠wt(f). The minimum k for which the graph G has an edge irregular total k-labeling is called the total irregularity streghth of the graph G. On this research we found that labeling algorithm of Cycle Chain Graph with n block cycle graph is an edge irregular total [(n-1)²+3(n-1+5/3]-labeling [ABSTRACT FROM AUTHOR]

A set is a neighbourhood total restrained dominating set of a graph if all the vertices in has one or more adjacent vertex in as well as in and also the each vertex in is adjacent to at least a vertex in and if the induced sub-graph has no isolated vertex. The cardinality of a minimum total restrained dominating set is called total restrained dominating number and is represented as. In this paper, we are introducing Neighborhood total restrained domination and develop an algorithm to find neighbourhood total restrained dominating set for Interval graphs and circular-arc graphs. [ABSTRACT FROM AUTHOR]

Installations of an Energy Storage System (ESS) with various functions such as power stabilization of renewable energy, demand management, and frequency adjustment are increasing. In particular, ESS for demand management is being established for high-voltage customers (300 KVA–1000 KVA) who have placed an Auto Section Switch (ASS) at the connection point within the distribution system. However, a power outage may occur in the Power Receiving System (PRS) when a short-circuit fault due to insulation breakdown occurs at the ESS DC side. The reason for this breakdown is that the fault current is reduced by transformer impedance, and the ASS is opened before the DC power fuse. Therefore, using the Graphic Solution Method (GSM), this paper presents an operation algorithm for protection coordination that isolates the fault section by first operating the DC power fuse with a small fault current. Furthermore, fault analysis modeling for a PRS composed of a switchgear section, a main distribution panel, a Power Conditioning System (PCS), a power fuse, and a battery is performed through PSCAD/EMTDC. From the simulation results, it is confirmed that the fault section is quickly isolated, and power outages for high-voltage customers are prevented because the DC power fuse selected by the proposed operation algorithm of protection coordination is opened before the ASS. [ABSTRACT FROM AUTHOR]

When can we compute the diameter of a graph in quasi linear time? We address this question for the class of split graphs, that we observe to be the hardest instances for deciding whether the diameter is at most two. We stress that although the diameter of a non-complete split graph can only be either 2 or 3, under the Strong ExponentialTime Hypothesis (SETH) we cannot compute the diameter of an n-vertex m-edge split graph in less than quadratic time – in the size n + m of the input. Therefore it is worth to study the complexity of diameter computation on subclasses of split graphs, in order to better understand the complexity border. Specifically, we consider the split graphs with bounded clique-interval number and their complements, with the former being a natural variation of the concept of interval number for split graphs that we introduce in this paper. We first discuss the relations between the clique-interval number and other graph invariants such as the classic interval number of graphs, the treewidth, the VC-dimension and the stabbing number of a related hypergraph. Then, in part based on these above relations, we almost completely settle the complexity of diameter computation on these subclasses of split graphs: • For the k-clique-interval split graphs, we can compute their diameter in truly subquadratic time if k = O(1), and even in quasi linear time if k = o(log n) and in addition a corresponding ordering of the vertices in the clique is given. However, under SETH this cannot be done in truly subquadratic time for any k = ω(log n). • For the complements of k-clique-interval split graphs, we can compute their diameter in truly subquadratic time if k = O(1), and even in time O(km) if a corresponding ordering of the vertices in the stable set is given. Again this latter result is optimal under SETH up to polylogarithmic factors. Our findings raise the question whether a k-clique interval ordering can always be computed in quasi linear time. We prove that it is the case for k = 1 and for some subclasses such as bounded-treewidth split graphs, threshold graphs and comparability split graphs. Finally, we prove that some important subclasses of split graphs – including the ones mentioned above – have a bounded clique-interval number. [ABSTRACT FROM AUTHOR]

An outer-1-planar graph is a graph admitting a drawing in the plane so that all vertices appear in the outer region of the drawing and every edge crosses at most one other edge. This paper establishes the local structure of outer-1- planar graphs by proving that each outer-1-planar graph contains one of the seventeen fixed configurations, and the list of those configurations is minimal in the sense that for each fixed configuration there exist outer-1-planar graphs containing this configuration that do not contain any of another sixteen configurations. There are two interesting applications of this structural theorem. First of all, we conclude that every (resp. maximal) outer-1-planar graph of minimum degree at least 2 has an edge with the sum of the degrees of its two end-vertices being at most 9 (resp. 7), and this upper bound is sharp. On the other hand, we show that the list 3-dynamic chromatic number of every outer-1-planar graph is at most 6, and this upper bound is best possible. [ABSTRACT FROM AUTHOR]

Modelling and predicting the suspect activity trajectory are of great importance for preventing and fighting crime in the food safety area. Combing artificial intelligence and the multiple U-model algorithm, this paper represents a novel approach to predict the suspect activity trajectory. Based on social text data, emotional assessment is conducted using the LSTM network to detect food safety criminal suspects. Activity trajectories of criminal suspects are clustered using the graphic clustering method based on the GPS data. U-model with the sliding window algorithm is proposed to model activity trajectories. Further, the multiple U-model strategy is proposed to predict the activity trajectory based on the accumulated model error of previous positions and multiple clustered trajectories. The simulation study shows that the proposed scheme can detect food safety criminal suspects and predict their activity trajectories effectively. [ABSTRACT FROM AUTHOR]

We present a general computer algorithm to contract an arbitrary number of second-quantized expressions and simplify the obtained analytical result. The functions that perform these operations are a part of the program Nostromo which facilitates the handling and analysis of the complicated mathematical formulas which are often encountered in modern quantum-chemical models. In contrast to existing codes of this kind, Nostromo is based solely on the Goldstone-diagrammatic representation of algebraic expressions in Fock space and has capabilities to work with operators as well as scalars. Each Goldstone diagram is internally represented by a line of text which is easy to interpret and transform. The calculation of matrix elements does not exploit Wick’s theorem in a direct way, but uses diagrammatic techniques to produce only nonzero terms. The identification of equivalent expressions and their subsequent factorization in the final result is performed easily by analyzing the topological structure of the diagrammatic expressions. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Opportunistic multihop networks with mobile relays recently have drawn much attention from researchers across the globe due to their wide applications in various challenging environments. However, because of their peculiar intrinsic features like lack of continuous connectivity, network partitioning, highly dynamic behavior, and long delays, it is very arduous to model and effectively capture the temporal variations of such networks with the help of classical graph models. In this work, we utilize an evolving graph to model the dynamic network and propose a matrix‐based algorithm to generate all minimal path sets between every node pair of such network. We show that these time‐stamped‐minimal‐path sets (TS‐MPS) between each given source‐destination node pair can be used, by utilizing the well‐known Sum‐of‐Disjoint Products technique, to generate various reliability metrics of dynamic networks, ie, two‐terminal reliability of dynamic network and its related metrics, ie, two‐terminal reliabilities of the foremost, shortest, and fastest TS‐MPS, and Expected Hop Count. We also introduce and compute a new network performance metric−Expected Slot Count. We use two illustrative examples of dynamic networks, one of four nodes, and the other of five nodes, to show the salient features of our technique to generate TS‐MPS and reliability metrics. [ABSTRACT FROM AUTHOR]

Motivation RNA secondary structure is a useful representation for studying the function of RNA, which captures most of the free energy of RNA folding. Using empirically determined energy parameters, secondary structures of nucleic acids can be efficiently computed by recursive algorithms. Several software packages supporting this task are readily available. As RNA secondary structures are outerplanar graphs, they can be drawn without intersection in the plane. Interpretation by the practitioner is eased when these drawings conform to a series of additional constraints beyond outerplanarity. These constraints are the reason why RNA drawing is difficult. Many RNA drawing algorithms therefore do not always produce intersection-free (outerplanar) drawings. Results To remedy this shortcoming we propose here the RNApuzzler algorithm which is guaranteed to produce intersection-free drawings. It is based on a drawing algorithm respecting constraints based on nucleotide distances (RNAturtle). We investigate relaxations of these constraints allowing for intersection-free drawings. Based on these relaxations, we implemented a fully automated, simple, and robust algorithm that produces aesthetic drawings adhering to previously established guidelines. We tested our algorithm using the RFAM database and found that we can compute intersection-free drawings of all RNAs therein efficiently. Availability and implementation The software can be accessed freely at: https://github.com/dwiegreffe/RNApuzzler. Supplementary information Supplementary data are available at Bioinformatics online. [ABSTRACT FROM AUTHOR]

Abstract Datasets are often collected from different resources or comprised of multiple representations (i.e., views). Multi-view clustering aims to analyze the multi-view data in an unsupervised way. Owing to the efficiency of uncovering the hidden structures of data, graph-based approaches have been investigated widely for various multi-view learning tasks. However, similarity measurement in these methods is challenging since the construction of similarity graph is impacted by several factors such as the scale of data, neighborhood size, choice of similarity metric, noise and outliers. Moreover, nonlinear relationships usually exist in real-world datasets, which have not been considered by most existing methods. In order to address these challenges, a novel model which simultaneously performs multi-view clustering task and learns similarity relationships in kernel spaces is proposed in this paper. The target optimal graph can be directly partitioned into exact c connected components if there are c clusters. Furthermore, our model can assign ideal weight for each view automatically without additional parameters as previous methods do. Since the performance is often sensitive to the input kernel matrix, the proposed model is further extended with multiple kernel learning ability. With the proposed joint model, three subtasks including construct the most accurate similarity graph, automatically allocate optimal weight for each view and find the cluster indicator matrix can be simultaneously accomplished. By this joint learning, each subtask can be mutually enhanced. Experimental results on benchmark datasets demonstrate that our model outperforms other state-of-the-art multi-view clustering algorithms. [ABSTRACT FROM AUTHOR]

Let p: X → X be a regular covering projection of connected graphs, where CTp denotes the group of covering transformations. Suppose that a group G ≤ Aut X lifts along p to a group G ≤ Aut X. The corresponding short exact sequence id → CTp → G → G → id is split sectional over a G-invariant subset of vertices Ω ⊆ V (X) if there exists a sectional complement, that is, a complement G to CTp with a G-invariant section Ω ⊂ V (X) over Ω. Such lifts do not split just abstractly but also permutationally in the sense that they enable a nice combinatorial description. Sectional complements are characterized from several viewpoints. The connection between the number of sectional complements and invariant sections on one side, and the structure of the split extension itself on the other, is analyzed. In the case when CTp is abelian and the covering projection is given implicitly in terms of a voltage assignment on the base graph X, an efficient algorithm for testing whether G has a sectional complement is presented. Efficiency resides on avoiding explicit reconstruction of the covering graph and the lifted group. The method extends to the case when CTp is solvable. [ABSTRACT FROM AUTHOR]

Abstract Given two graphs, the graph matching problem is to align the two vertex sets so as to minimize the number of adjacency disagreements between the two graphs. The seeded graph matching problem is the graph matching problem when we are first given a partial alignment that we are tasked with completing. In this article, we modify the state-of-the-art approximate graph matching algorithm "FAQ" of Vogelstein et al. (2015) to make it a fast approximate seeded graph matching algorithm, adapt its applicability to include graphs with differently sized vertex sets, and extend the algorithm so as to provide, for each individual vertex, a nomination list of likely matches. We demonstrate the effectiveness of our algorithm via simulation and real data experiments; indeed, knowledge of even a few seeds can be extremely effective when our seeded graph matching algorithm is used to recover a naturally existing alignment that is only partially observed. [ABSTRACT FROM AUTHOR]

This paper investigates the vertex coloring problem in an uncertain graph in which all vertices are deterministic, while all edges are not deterministic and exist with some degree of belief in uncertain measures. The concept of the maximal uncertain independent vertex set of an uncertain graph is first introduced. We then present a degree of belief rule to obtain the family of maximal uncertain independent vertex sets. Based on the maximal uncertain independent vertex set, some properties of the separation degree of an uncertain graph are discussed. Following that, the concept of an uncertain chromatic set is introduced. Then, a maximum separation degree algorithm is derived to obtain the uncertain chromatic set. Finally, numerical examples are presented to demonstrate the application of the vertex coloring problem in uncertain graphs and the effectiveness of the maximum separation degree algorithm. [ABSTRACT FROM AUTHOR]

Abstract Motivated by the study of genomes evolving by reversals, we consider pseudograph transformations known as "pressing sequences". In particular, we address the question of when a graph has precisely one pressing sequence resulting in the empty graph, thus answering an question from Cooper and Davis (2015) [13]. We characterize such "uniquely pressable" graphs, count the number of them on a given number of vertices, and provide a polynomial-time recognition algorithm. We conclude with several open questions. [ABSTRACT FROM AUTHOR]

Abstract In this work we compare solutions of two distinct algorithms that try to increase the value of the algebraic connectivity of a given graph by choosing, with different strategies, edges to be included. Eccentricity and Perturbation Heuristics were considered, as well as random graphs and special families of trees being inputs of those algorithms. As conclusions of the reported experiments, the Eccentricity Heuristic obtains good results when compared with Perturbation Heuristic and a conjecture about broom trees is presented. [ABSTRACT FROM AUTHOR]

Abstract The units forms are algebraic expressions that have important role in representation theory of algebras. We identified that existing algorithms have exponential time complexity for weakly nonnegative and weakly positive types. In this paper we introduce a polynomial algorithm for the recognition of weakly nonnegative unit forms. The related algorithm identifies hypercritical restrictions in a given unit form, testing every subgraph of 9 vertices of the unit form associated graph. By adding Depth First Search approach, a similar strategy could be used in the recognition of weakly positive unit forms. We also present the most popular methods to decide whether or not a unit form is weakly nonnegative or weakly positive, we analyze their time complexity and we compare the results with our algorithms. [ABSTRACT FROM AUTHOR]

Abstract A set D ⊆ V of a graph G = (V , E) is called a dominating set of G if every vertex in V ∖ D has at least one neighbor in D. A dominating set D of G is a paired-dominating set of G if G [ D ] , the subgraph of G induced by D , has a perfect matching. Given a graph G , Min-Paired-Dom-Set is the problem to find a paired-dominating set of G of minimum cardinality. Min-Paired-Dom-Set is known to be NP -hard for general graphs and many other restricted classes of graphs. However, Min-Paired-Dom-Set is solvable in polynomial time in some graph classes including strongly chordal graphs and chordal bipartite graphs. In this paper, we strengthen this result by proposing a polynomial time algorithm to compute a minimum paired-dominating set in the class of strongly orderable graphs, which includes strongly chordal graphs and chordal bipartite graphs. The algorithm runs in linear time if a quasi-simple elimination ordering of the strongly orderable graph is provided. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we empirically profile three algorithms for the Shortest Negative Cost Cycle (SNCC) problem. In this problem, we are given a directed, weighted graph G = 〈 V , E , c 〉 , and the goal is to identify a negative cost cycle with the fewest number of edges. The SNCC problem finds applications in a number of domains ranging from program verification to certifying algorithm design. The first polynomial time algorithm for this problem was proposed in Subramani (2009). Since then, several additional techniques, including a randomized approach, have been proposed for the SNCC problem. Our goal in this paper is to analyze three algorithmic paradigms for this problem from the empirical perspective. Towards this end, we profile these algorithmic approaches over a wide range of inputs. Our results indicated that the randomized algorithm is the algorithm of choice over a number of graph classes. [ABSTRACT FROM AUTHOR]

Abstract The paper is devoted to a Gram classification of an important class of signed graphs with loops playing an important role in many branches of mathematics, physics and computer science including game theory, crystallography, Diophantine geometry, Lie theory, spectral graph theory, a graph coloring technique, algebraic methods in graph theory. They have many deep applications in spectral machine learning methods, electrical networks, to link sign prediction in signed unipartite and bipartite networks, and community partition in social networks. We continue the study of finite connected edge-bipartite graphs Δ (bigraphs, for short), with m ≥ 2 vertices (a class of signed graphs), started in Simson (2013) and developed in Simson (2016) by means of the non-symmetric Gram matrix G ˇ Δ ∈ M m (Z) defining Δ , its symmetric Gram matrix G Δ : = 1 2 [ G ˇ Δ + G ˇ Δ t r ] ∈ M m (1 2 Z) , and the Gram quadratic form q Δ : Z m → Z. In the present paper, given n ≥ 1 , we mainly study connected principal Cox-regular edge-bipartite graphs Δ with loops and n + 1 ≥ 2 vertices, in the sense that the symmetric Gram matrix G Δ ∈ M n + 1 (Z) of Δ is positive semi-definite of rank n ≥ 1. Our aim is to classify such edge-bipartite graphs by means of an inflation algorithm, up to the weak Gram Z -congruence Δ ∼ Z Δ ′ , where Δ ∼ Z Δ ′ means that G Δ ′ = B t r ⋅ G Δ ⋅ B , for some B ∈ M n + 1 (Z) with det B = ± 1. Our main result of the paper asserts that, given a principal Cox-regular connected edge-bipartite graph Δ with n + 1 ≥ 2 vertices, there exists an edge-bipartite Euclidean graph D n + 1 of Tables A and B, and a suitably chosen sequence t • − of the inflation operators, each of one of the forms Δ ′ ↦ t a − Δ ′ or Δ ′ ↦ t a b − Δ ′ , such that the composite operator Δ ↦ t • − Δ reduces Δ to the bigraph D n + 1 such that: (i) Δ ∼ Z D n + 1 and (ii) the bigraphs Δ , and D n + 1 have the same number of loops. The algorithm does not change loops and computes a matrix B ∈ M n + 1 (Z) , with det B = ± 1 , defining the weak Gram Z -congruence Δ ∼ Z D n + 1 , that is, satisfying the equation G D n + 1 = B t r ⋅ G Δ ⋅ B. [ABSTRACT FROM AUTHOR]

Highlights • We present a novel view-based 3D object retrieval framework, which is deployed over a graph-based collaborative learning scheme to intelligently fuse multiple features. • The hypergraph based collaborative feature learning scheme to fuse complement descriptors from both the contour and the interior region of 3D object effectively. • The view-based 3D object retrieval is done via a greedy bipartite graph matching algorithm, which achieves highly accurate and efficient 3D object matching. • The interactive learning procedure with user feedbacks achieved better performance. Abstract 3D object retrieval has attractive extensive research focus in recent years. Among various schemes, view based 3D object retrieval is regarded as a promising direction. In this paper, we present a novel view-based 3D object retrieval framework, which is deployed over a graph-based collaborative learning scheme to intelligently fuse multiple features. In particular, we introduce a hypergraph based collaborative feature learning scheme to fuse complement descriptors from both the contour and the interior region of 3D object effectively. Then, the view-based 3D object retrieval is done via a greedy bipartite graph matching algorithm, which achieves highly accurate and efficient 3D object matching. With the above bipartite graph matching and feature concatenation, significant performance improvement is achieved in the 3D object retrieval task, on either widely-used benchmark datasets or open competitions like SHREC15 challenge. In both evaluations, the proposed graph-based collaborative feature learning scheme has beaten a serial of existing approaches and state-of-the-art schemes. [ABSTRACT FROM AUTHOR]

Representation of any network graphically has vast applications and used for knowledge extraction efficiently. Due to the increase in applications of a graph, the size of the graph becomes larger as well as its complexity becomes more and more. So visualization and analyzing of a large community graph are more challenging. Hence compression technique may be used to study a large community graph for knowledge extraction. During compression, there should not be any loss of information. This paper proposes an algorithm, "ComComGra" which compresses a large community graph with various communities using graph mining techniques. The proposed algorithm elaborates with two examples which include a benchmark example. [ABSTRACT FROM AUTHOR]

Graph matching aims at establishing correspondences between graph elements, and is widely used in many computer vision tasks. Among recently proposed graph matching algorithms, those utilizing the path following strategy have attracted special research attentions due to their exhibition of state-of-the-art performances. However, the paths computed in these algorithms often contain singular points, which could hurt the matching performance if not dealt properly. To deal with this issue, we propose a novel path following strategy, named branching path following (BPF), to improve graph matching accuracy. In particular, we first propose a singular point detector by solving a KKT system, and then design a branch switching method to seek for better paths at singular points. Moreover, to reduce the computational burden of the BPF strategy, an adaptive path estimation (APE) strategy is integrated into BPF to accelerate the convergence of searching along each path. A new graph matching algorithm named ABPF-G is developed by applying APE and BPF to a recently proposed path following algorithm named GNCCP (Liu & Qiao 2014). Experimental results reveal how our approach consistently outperforms state-of-the-art algorithms for graph matching on five public benchmark datasets. [ABSTRACT FROM AUTHOR]

We consider a statistical model for the problem of finding subgraphs with specified topology in an otherwise random graph. This task plays an important role in the analysis of social and biological networks. In these types of networks, small subgraphs with a specific structure have important functional roles, and they are referred to as motifs. Within this model, one or multiple copies of a subgraph is added (planted) in an Erdős–Renyi random graph with $n$ vertices and edge probability $q_{0}$. We ask whether the resulting graph can be distinguished reliably from a pure Erdős–Renyi random graph, and we present two types of result. First we investigate the question from a purely statistical perspective, and ask whether there isanytest that can distinguish between the two graph models. We provide necessary and sufficient conditions that are essentially tight for small enough subgraphs. Next we study two polynomial-time algorithms for solving the same problem: a spectral algorithm and a semidefinite programming (SDP) relaxation. For the spectral algorithm, we establish sufficient conditions under which it distinguishes the two graph models with high probability. Under the same conditions the spectral algorithm indeed identifies the hidden subgraph. The spectral algorithm is substantially sub-optimal with respect to the optimal test. We show that a similar gap is present for the more sophisticated SDP approach. [ABSTRACT FROM AUTHOR]

In this paper, we consider the discrete unit square cover (DUSC) problem as follows: given a set 𝒫 of n points and a set 𝒮 of m axis-aligned unit squares in ℝ 2 , the objective is (i) to check whether the union of the squares in 𝒮 covers all the points in 𝒫 , and (ii) if the answer is yes, then select a minimum cardinality subset 𝒮 ∗ ⊆ 𝒮 such that each point in 𝒫 is covered by at least one square in 𝒮 ∗ . For the DUSC problem: (i) we propose a (2 + 4 k − 2) -approximation algorithm, where k (> 2) is an integer parameter that defines a trade-off between the running time and the approximation factor of the algorithm. The running time of our proposed algorithm is O (k m k n + n log n). Our solution of the DUSC problem is based on a simple (1 + 2 k − 2) -approximation algorithm for the subproblem strip square cover (SSC) problem, where all the points in 𝒫 are lying within a horizontal strip of unit height. (ii) we also propose a 2-approximation algorithm, which runs in O (m 4 n) time. The 2-approximation algorithm is based on an algorithm for the subproblem SSC problem. The algorithm for the subproblem is developed using plane sweep and graph search traversal techniques. We also extend this algorithm to get 2-approximation result for the weighted DUSC problem where the squares are assigned weights, and the aim is to choose a subset 𝒮 ∗ ⊆ 𝒮 such that each point in 𝒫 is covered by at least one square in 𝒮 ∗ and the sum of the weights of squares in 𝒮 ∗ is minimized. [ABSTRACT FROM AUTHOR]

A star edge coloring of a graph G is a proper edge coloring without bichromatic paths and cycles of length four. In this paper, we obtain the star edge chromatic number of the corona product of path with cycle, path with wheel, path with helm and path with gear graphs, denoted by Pm ◦ Cn, Pm ◦ Wn, Pm ◦ Hn, Pm ◦ Gn respectively. [ABSTRACT FROM AUTHOR]

A Not-All-Equal decomposition of a graph G is a decomposition of the vertices of G into two parts such that each vertex in G has at least one neighbor in each part. Also, a 1-in-Degree decomposition of a graph G is a decomposition of the vertices of G into two parts A and B such that each vertex in the graph G has exactly one neighbor in part A. Among our results, we show that for a given graph G, if G does not have any cycle of length congruent to 2 mod 4, then there is a polynomial time algorithm to decide whether G has a 1-in-Degree decomposition. In sharp contrast, we prove that for every r, r≥3, for a given r-regular bipartite graph G determining whether G has a 1-in-Degree decomposition is NP-complete. These complexity results have been especially useful in proving NP-completeness of various graph related problems for restricted classes of graphs. In consequence of these results we show that for a given bipartite 3-regular graph G determining whether there is a vector in the null-space of the 0,1-adjacency matrix of G such that its entries belong to {±1,±2} is NP-complete. Among other results, we introduce a new version of Planar 1-in-3 SAT and we prove that this version is also NP-complete. In consequence of this result, we show that for a given planar (3, 4)-semiregular graph G determining whether there is a vector in the null-space of the 0,1-incidence matrix of G such that its entries belong to {±1,±2} is NP-complete. [ABSTRACT FROM AUTHOR]

Abstract Let P be a path graph of n vertices embedded in a metric space. We consider the problem of adding a new edge to P such that the diameter of the resulting graph is minimized. Previously (in ICALP 2015) the problem was solved in O (n log 3 ⁡ n) time. In this paper, based on new observations and different algorithmic techniques, we present an O (n log ⁡ n) time algorithm. [ABSTRACT FROM AUTHOR]

A generalized lollipop graph is formed by connecting a tree and a threshold graph with an edge. Motivated by a sequence of algorithms that compute the characteristic polynomial of some classes of graphs, we present an algorithm for computing the characteristic polynomial of generalized lollipop graph with relation to signless Laplacian matrix Q . As application, we show how to construct graphs having Q -cospectral mate. [ABSTRACT FROM AUTHOR]

This article proposes the tolerance nearness measure (TNM) as a computationally reduced alternative to the graph edit distance (GED) for performing graph comparisons. The TNM is defined within the context of near set theory, where the central idea is that determining similarity between sets of disjoint objects is at once intuitive and practically applicable. The TNM between two graphs is produced using the Bron-Kerbosh maximal clique enumeration algorithm. The result is that the TNM approach is less computationally complex than the bipartite-based GED algorithm. The contribution of this paper is the application of TNM to the problem of quantifying the similarity of disjoint graphs and that the maximal clique enumeration-based TNM produces comparable results to the GED when applied to the problem of content-based image processing, which becomes important as the number of nodes in a graph increases. [ABSTRACT FROM AUTHOR]

We study the Unsplittable Flow problem (UFP) on trees with a submodular objective function. The input to this problem is a tree with edge capacities and a collection of tasks, each characterized by a source node, a sink node, and a demand. A subset of the tasks is feasible if the tasks can simultaneously send their demands from the source to the sink without violating the edge capacities. The goal is to select a feasible subset of the tasks that maximizes a submodular objective function. Our main result is an O(klogn)-approximation algorithm for Submodular UFP on trees where k denotes the pathwidth of the given tree. Since every tree has pathwidth O(logn), we obtain an O(log2n) approximation for arbitrary trees. This is the first non-trivial approximation guarantee for the problem, matching the best known approximation ratio for UFP on trees with a linear objective function. Our main technical contribution is a new geometric relaxation for UFP on trees that builds on the recent work of Bonsma et al. (2014, FOCS), Anagnostopoulos et al. (Amazing 2+ϵ approximation for unsplittable flow on a path, SIAM, pp 26-41, 2014) for UFP on paths with a linear objective. Our relaxation is very structured, so it can be combined with the contention resolution framework of Chekuri et al. (2009, STOC). Our approach is robust and extends to several related problems, such as UFP with bag constraints and the Storage Allocation Problem. Additionally, we study the special case of UFP on trees with a linear objective and upward instances where, for each task, the source node is a descendant of the sink node. Such instances generalize UFP on paths. We build on the work of Bansal et al. (A quasi-PTAS for unsplittable flow on line graphs, ACM, pp 721-729, 2006) for UFP on paths and obtain a QPTAS for upward instances when the input data is quasi-polynomially bounded. We complement this result by showing that, unlike the path setting, upward instances are APX-hard if the input data is arbitrary. [ABSTRACT FROM AUTHOR]

We derive a simple bijection between geometric plane perfect matchings on 2n points in convex position and triangulations on n+2 points in convex position. We then extend this bijection to monochromatic plane perfect matchings on periodically k-colored vertices and (k+2)-gonal tilings of convex point sets. These structures are related to a generalization of Temperley-Lieb algebras and our bijections provide explicit one-to-one relations between matchings and tilings. Moreover, for a given element of one class, the corresponding element of the other class can be computed in linear time. [ABSTRACT FROM AUTHOR]

We solve the subgroup intersection problem (SIP) for any RAAG G of Droms type (i.e. with defining graph not containing induced squares or paths of length 3): there is an algorithm which, given finite sets of generators for two subgroups H , K ≤ G , decides whether H ∩ K is finitely generated or not, and, in the affirmative case, it computes a set of generators for H ∩ K. Taking advantage of the recursive characterization of Droms groups, the proof consists in separately showing that the solvability of SIP passes through free products, and through direct products with free-abelian groups. We note that most of RAAGs are not Howson, and many (e.g. 𝔽 2 × 𝔽 2 ) even have unsolvable SIP. [ABSTRACT FROM AUTHOR]

Consider a simple locally finite hypergraph on a countable vertex set, where each edge represents one unit of load which should be distributed among the vertices defining the edge. An allocation of load is called balanced if load cannot be moved from a vertex to another that is carrying less load. We analyze the properties of balanced allocations of load. We extend the concept of balancedness from finite hypergraphs to their local weak limits in the sense of Benjamini and Schramm (Electron J Probab 6(23):13, 2001) and Aldous and Steele (in: Probability on discrete structures. Springer, Berlin, pp 1-72, 2004). To do this, we define a notion of unimodularity for hypergraphs which could be considered an extension of unimodularity in graphs. We give a variational formula for the balanced load distribution and, in particular, we characterize it in the special case of unimodular hypergraph Galton-Watson processes. Moreover, we prove the convergence of the maximum load under some conditions. Our work is an extension to hypergraphs of Anantharam and Salez (Ann Appl Probab 26(1):305-327, 2016), which considered load balancing in graphs, and is aimed at more comprehensively resolving conjectures of Hajek (IEEE Trans Inf Theory 36(6):1398-1414, 1990). [ABSTRACT FROM AUTHOR]

Abstract A graph of treewidth k has a representation by subtrees of a ternary tree, with subtrees of adjacent vertices sharing a tree node, and any tree node sharing at most k + 1 subtrees. Likewise for branchwidth, but with a shift to the edges of the tree rather than the nodes. In this paper we show that the mm-width of a graph – maximum matching width – combines aspects of both these representations, targeting tree nodes for adjacency and tree edges for the parameter value. The proof of this new characterization of mm-width is based on a definition of canonical minimum vertex covers of bipartite graphs. We show that these behave in a monotone way along branch decompositions over the vertex set of a graph. We use these representations to compare mm-width with treewidth and branchwidth, and also to give another new characterization of mm-width, by subgraphs of chordal graphs. We prove that given a graph G and a branch decomposition of maximum matching width k we can solve the Minimum Dominating Set Problem in time O ∗ (8 k) , thereby beating O ∗ (3 tw (G) ) whenever tw (G) > log 3 8 × k ≈ 1. 893 k. Note that mmw (G) ≤ tw (G) + 1 ≤ 3 mmw (G) and these inequalities are tight. Given only the graph G and using the best known algorithms to find decompositions, maximum matching width will be better for Minimum Dominating Set whenever tw (G) > 1. 549 × mmw (G). [ABSTRACT FROM AUTHOR]

Abstract Circle graphs are intersection graphs of chords in a circle and k -polygon graphs are intersection graphs of chords in a convex k -sided polygon where each chord has its endpoints on distinct sides. The k -polygon graphs, for k ≥ 2 , form an infinite chain of graph classes, each of which contains the class of permutation graphs. The union of all of those graph classes is the class of circle graphs. The polygon number ψ (G) of a circle graph G is the minimum k such that G is a k -polygon graph. Given a circle graph G and an integer k , determining whether ψ (G) ≤ k is NP-complete, while the problem is solvable in polynomial time for fixed k. In this paper, we show that ψ (G) is always at least as large as the asteroidal number of G , and equal to the asteroidal number of G when G is a connected distance hereditary graph that is not a clique. This implies that the classes of distance hereditary permutation graphs and distance hereditary AT-free graphs are the same, and we give a forbidden subgraph characterization of that class. We also establish the following upper bounds: ψ (G) is at most the clique cover number of G if G is not a clique, at most 1 plus the independence number of G , and at most ⌈ n ∕ 2 ⌉ where n ≥ 3 is the number of vertices of G. Our results lead to linear time algorithms for finding the minimum number of corners that must be added to a given circle representation to produce a polygon representation, and for finding the asteroidal number of a distance hereditary graph, both of which are improvements over previous algorithms for those problems. [ABSTRACT FROM AUTHOR]

Abstract We consider the Weak Roman Domination problem. Given an undirected graph G = (V , E) , the aim is to find a weak Roman domination function (wrd-function for short) of minimum cost, i.e. a function f : V → { 0 , 1 , 2 } such that every vertex v ∈ V is defended (i.e. there exists a neighbor u of v , possibly u = v , such that f (u) ⩾ 1) and for every vertex v ∈ V with f (v) = 0 there exists a neighbor u of v such that f (u) ⩾ 1 and the function f u → v defined by f u → v (v) = 1 , f u → v (u) = f (u) − 1 and f u → v (x) = f (x) otherwise does not contain any undefended vertex. The cost of a wrd-function f is defined by c o s t (f) = ∑ v ∈ V f (v). The trivial enumeration algorithm runs in time O ∗ (3 n) and polynomial space and is the best one known for the problem so far. We are breaking the trivial enumeration barrier by providing two faster algorithms: we first prove that the problem can be solved in O ∗ (2 n) time needing exponential space , and then describe an O ∗ (2. 227 9 n) algorithm using polynomial space. Our results rely on structural properties of a wrd-function, as well as on the best polynomial space algorithm for the Red-Blue Dominating Set problem. Moreover we show that the problem can be solved in linear-time on interval graphs. [ABSTRACT FROM AUTHOR]

The Wiener polarity index (usually denoted by W p ) of a graph G is defined as the number of unordered pairs of the vertices of G which are at distance 3. Denote by CT n the family of all n -vertex chemical trees. In a recent paper, Ashrafi and Ghalavand [1] determined the first three minimum W p values of n -vertex chemical trees for n  ≥ 7 and characterized the chemical trees attaining the first two minimum W p values among all the members of CT n for n  ≥ 4. In this note, the chemical trees with the third minimum W p value are characterized from the graph family CT n for n  ≥ 7, and the chemical trees from the family CT n , n  ≥ 4, with the first two minimum W p values are also obtained in an alternative but shorter way. [ABSTRACT FROM AUTHOR]

Background: Networks whose nodes have labels can seem complex. Fortunately, many have substructures that occur often (“motifs”). A societal example of a motif might be a household. Replacing such motifs by named supernodes reduces the complexity of the network and can bring out insightful features. Doing so repeatedly may give hints about higher level structures of the network. We call this recursive process Recursive Supernode Extraction. Results: This paper describes algorithms and a tool to discover disjoint (i.e. non-overlapping) motifs in a network, replacing those motifs by new nodes, and then recursing. We show applications in food-web and protein-protein interaction (PPI) networks where our methods reduce the complexity of the network and yield insights. Conclusions: SuperNoder is a web-based and standalone tool which enables the simplification of big graphs based on the reduction of high frequency motifs. It applies various strategies for identifying disjoint motifs with the goal of enhancing the understandability of networks. [ABSTRACT FROM AUTHOR]

Given a graph G = ( V , E ) of order n and maximum degree Δ , the NP -complete S - labeling problem consists in finding a labeling of G , i.e. a bijective mapping ϕ : V → { 1 , 2 … n } , such that SL ϕ ( G ) = ∑ u v ∈ E min { ϕ ( u ) , ϕ ( v ) } is minimized. In this paper, we study the S - labeling problem, with a particular focus on algorithmic issues. We first give intrinsic properties of optimal labelings, which will prove useful for our algorithmic study. We then provide lower bounds on SL ϕ ( G ) , together with a generic greedy algorithm, which collectively allow us to approximate the problem in several classes of graphs—in particular, we obtain constant approximation ratios for regular graphs and bounded degree graphs. We also show that deciding whether there exists a labeling ϕ of G such that SL ϕ ( G ) ≤ | E | + k is solvable in O ∗ ( 2 2 k ( 2 k ) ! ) time, thus fixed-parameterized tractable in k . We finally show that the S - Labeling problem is polynomial-time solvable for two classes of graphs, namely split graphs and (sets of) caterpillars. [ABSTRACT FROM AUTHOR]

An orientation of an undirected graph is obtained by assigning a direction to each of its edges. It is called cyclic when a directed cycle appears, and acyclic otherwise. We study efficient algorithms for enumerating the orientations of an undirected graph. To get the full picture, we consider both the cases of acyclic and cyclic orientations, under some rules specifying which nodes are the sources (i.e. their incident edges are all directed outwards). Our enumeration algorithms use linear space and provide new bounds for the delay, which is the maximum elapsed time between the output of any two consecutively listed solutions. We obtain a delay of O ( m ) for acyclic orientations and O ̃ ( m ) for cyclic ones. When just a single source is specified, these delays become O ( m ⋅ n ) and O ( m ⋅ h + h 3 ) , respectively, where h is the girth of the graph without the given source. When multiple sources are specified, the delays are the same as in the single source case. [ABSTRACT FROM AUTHOR]

Canonical correlation analysis(CCA) and its variants have been playing an important role in dimension reduction and information fusion. Most of the existing CCA-related algorithms can only handle fully-paired scenarios, where all the samples from different views require to be rigorously paired. However, majority of the samples only provide partial pairwise correspondences in most cases. In such semi-paired scenarios, traditional methods often perform poorly due to overfitting. Furthermore, most CCA-related algorithms only focus on the correlation between different views but ignore preserving the within-view similarity which is helpful to make samples in the same class mapped more compactly onto canonical correlation space. In addition, since CCA is unable to exploit discriminative information effectively, improved performance is limited in many tasks. To solve these challenging problems, we propose a novel method named joint Intermodal and Intramodal Semi-paired Canonical Correlation Analysis (I 2 SCCA), which properly handle the semi-paired scenarios. At the same time, the method preserves the within-view similarity information by making two neighbor samples in the same view closer in the projected space. Different from traditional similarity matrix construction, we incorporate a rank constraint of Laplacian matrix into the graph-based clustering algorithms. With fewer parameters, a graph that has exactly c (the number of clusters) connected components is learned without requiring any post graph-cut processing. Based on the obtained graph, we estimate the cross-view correlation through similarity measure of the single view. Specially, taking advantage of clustering assumption, discriminative information is derived although there is no label available. Finally, we extend I 2 SCCA to multi-view and non-linear cases. Extensive experiments on several datasets demonstrate the effectiveness of I 2 SCCA. [ABSTRACT FROM AUTHOR]

Light field cameras capture the 3D information in a scene with a single exposure. This special feature makes light field cameras very appealing for a variety of applications: from post-capture refocus to depth estimation and image-based rendering. However, light field cameras suffer by design from strong limitations in their spatial resolution. Off-the-shelf super-resolution algorithms are not ideal for light field data, as they do not consider its structure. On the other hand, the few super-resolution algorithms explicitly tailored for light field data exhibit significant limitations, such as the need to carry out a costly disparity estimation procedure with sub-pixel precision. We propose a new light field super-resolution algorithm meant to address these limitations. We use the complementary information in the different light field views to augment the spatial resolution of the whole light field at once. In particular, we show that coupling the multi-view approach with a graph-based regularizer, which enforces the light field geometric structure, permits to avoid the need of a precise and costly disparity estimation step. Extensive experiments show that the new algorithm compares favorably to the state-of-the-art methods for light field super-resolution, both in terms of visual quality and in terms of reconstruction error. [ABSTRACT FROM AUTHOR]

Let G be a graph such that each vertex has its list of available colors, and assume that each list is a subset of the common set consisting of k colors. For two given list colorings of G , we study the problem of transforming one into the other by changing only one vertex color assignment at a time, while at all times maintaining a list coloring. This problem is known to be PSPACE-complete even for bounded bandwidth graphs and a fixed constant k . In this paper, we study the fixed-parameter tractability of the problem when parameterized by several graph parameters. We first give a fixed-parameter algorithm for the problem when parameterized by k and the modular-width of an input graph. We next give a fixed-parameter algorithm for the shortest variant which computes the length of a shortest transformation when parameterized by k and the size of a minimum vertex cover of an input graph. As corollaries of these two results, we show that the problem for cographs and the shortest variant for split graphs are fixed-parameter tractable even when only k is taken as a parameter. On the other hand, we prove that the problem is W [ 1 ] -hard when parameterized only by the size of a minimum vertex cover of an input graph. [ABSTRACT FROM AUTHOR]

A graph is said to be of class 1 if its edge chromatic number is equal to the maximum degree of this graph. Let G be a planar graph with maximum degree Δ ≥ 6 and without adjacent 7-cycles, then G is of class 1. [ABSTRACT FROM AUTHOR]

The quest for colorful components (connected components where each color is associated with at most one vertex) inside a vertex-colored graph has been widely considered in the last ten years. Here we consider two variants, Minimum Colorful Components (MCC) and Maximum Edges in transitive Closure (MEC), introduced in 2011 in the context of orthology gene identification in bioinformatics. The input of both MCC and MEC is a vertex-colored graph. MCC asks for the removal of a subset of edges, so that the resulting graph is partitioned in the minimum number of colorful connected components; MEC asks for the removal of a subset of edges, so that the resulting graph is partitioned in colorful connected components and the number of edges in the transitive closure of such a graph is maximized. We study the parameterized and approximation complexity of MCC and MEC, for general and restricted instances. For MCC on trees we show that the problem is basically equivalent to Minimum Cut on Trees, thus MCC is not approximable within factor 1.36 − ε , it is fixed-parameter tractable and it admits a poly-kernel (when the parameter is the number of colorful components). Moreover, we show that MCC, while it is polynomial time solvable on paths, it is NP-hard even for graphs with constant distance to disjoint paths number. Then we consider the parameterized complexity of MEC when parameterized by the number k of edges in the transitive closure of a solution (the graph obtained by removing edges so that it is partitioned in colorful connected components). We give a fixed-parameter algorithm for MEC parameterized by k and, when the input graph is a tree, we give a poly-kernel. [ABSTRACT FROM AUTHOR]

We provide a combinatorial algorithm for constructing the stable Auslander–Reiten component containing a given indecomposable module of a symmetric special biserial algebra using only information from its underlying Brauer graph. We also show that the structure of the Auslander–Reiten quiver is closely related to the distinct Green walks of the Brauer graph and detail the relationship between the precise shape of the stable Auslander–Reiten components for domestic Brauer graph algebras and their underlying graph. Furthermore, we show that the specific component containing a given simple or indecomposable projective module for any Brauer graph algebra is determined by the edge in the Brauer graph associated to the module. [ABSTRACT FROM AUTHOR]

Highlights • Development of a dynamic multi-disciplinary repository based on semantics. • Development of an ontological representation of the biorefinery pathways. • Algorithms for the synthesis of value chains for selected feedstock or/and product based on the structure of the ontology and semantic queries. • The use of semantics contributes to the creation of new knowledge from the ontology. Abstract There is an increased use of problem representations (i.e. superstructures in synthesis problems; networks in route problems; graphs; ordered graphs in various systems representations) following on significant advances in optimization technologies that hold capabilities to solve, robustly, large-scale problems. In an attempt to systematically tackle disparate domains and build high-throughput functions, the paper contributes with a semantically-enabled approach systematized and engineered by ontologies. The aim is to develop an intelligent environment with capabilities to build and scale-up system representations, automatically. The work is demonstrated on problems akin to biorenewables and biorefineries; an identical approach is possible to the general problem. Using relations and rules defined among entities, semantics are deployed to model and expand domains (biorefinery pathways) whereas enabling extracting and creating knowledge. The repository, already on a web-based platform and available as open-source, essentially upgrades conventional representations with capabilities to share (import/export) and integrate its content externally. [ABSTRACT FROM AUTHOR]

We prove a general large-sieve statement in the context of random walks on subgraphs of a given graph. This can be seen as a generalization of previously known results where one performs a random walk on a group enjoying a strong spectral gap property. In such a context the point is to exhibit a strong uniform expansion property for a suitable family of Cayley graphs on quotients. In our combinatorial approach, this is replaced by a result of Alon–Roichman about expanding properties of random Cayley graphs. Applying the general setting we show, for instance, that with high probability (in a strong explicit sense) random coloured subsets of integers contain monochromatic (nonempty) subsets summing to $0$ , and that a random colouring of the edges of a complete graph contains a monochromatic triangle. [ABSTRACT FROM AUTHOR]

We show that any regular pseudocomplemented Kleene algebra defined on an algebraic lattice is isomorphic to a rough set Kleene algebra determined by a tolerance induced by an irredundant covering. [ABSTRACT FROM AUTHOR]