Restricted arc-connectivity of digraphs

Since interconnection networks are often modeled by graphs or digraphs, the edge-connectivity of a graph or arc-connectivity of a digraph are important measurements for fault tolerance of networks.The restricted edge-connectivity λ^′(G) of a graph G is the minimum cardinality over all edge-cuts S in a graph G such that there are no isolated vertices in G−S. A connected graph G is called λ^′-connected, if λ^′(G) exists.In 1988, Esfahanian and Hakimi [A.H. Esfahanian, S.L. Hakimi, On computing a conditional edge-connectivity of a graph, Inform. Process. Lett. 27 (1988), 195-199] have shown that each connected graph G of order n?4, except a star, is λ^′-connected and satisfies λ^′(G)?ξ(G), where ξ(G) is the minimum edge-degree of G.If D is a strongly connected digraph, then we call in this paper an arc set S a restricted arc-cut of D if D−S has a non-trivial strong component D₁ such that D−V(D₁) contains an arc. The restricted arc-connectivity λ^′(D) is the minimum cardinality over all restricted arc-cuts S.We observe that the recognition problem, whether λ^′(D) exists for a strongly connected digraph D is solvable in polynomial time. Furthermore, we present some analogous results to the above mentioned theorem of Esfahanian and Hakimi for digraphs, and we show that this theorem follows easily from one of our results.     

Lindex: a lattice-based index for graph databases

Subgraph querying has wide applications in various fields such as cheminformatics and bioinformatics. Given a query graph, q, a subgraph-querying algorithm retrieves all graphs, D(q), which have q as a subgraph, from a graph database, D. Subgraph querying is costly because it uses subgraph isomorphism tests, which are NP-complete. Graph indices are commonly used to improve the performance of subgraph querying in graph databases. Subgraph-querying algorithms first construct a candidate answer set by filtering out a set of false answers and then verify each candidate graph using subgraph isomorphism tests. To build graph indices, various kinds of substructure (subgraph, subtree, or path) features have been proposed with the goal of maximizing the filtering rate. Each of them works with a specifically designed index structure, for example, discriminative and frequent subgraph features work with gIndex, δ-TCFG features work with FG-index, etc. We propose Lindex, a graph index, which indexes subgraphs contained in database graphs. Nodes in Lindex represent key-value pairs where the key is a subgraph in a database and the value is a list of database graphs containing the key. We propose two heuristics that are used in the construction of Lindex that allows us to determine answers to subgraph queries conducting less subgraph isomorphism tests. Consequently, Lindex improves subgraph-querying efficiency. In addition, Lindex is compatible with any choice of features. Empirically, we demonstrate that Lindex used in conjunction with subgraph indexing features proposed in previous works outperforms other specifically designed index structures. As a novel index structure, Lindex (1) is effective in filtering false graphs (2) provides fast index lookups, (3) is fast with respect to index construction and maintenance, and (4) can be constructed using any set of substructure index features. These four properties result in a fast and scalable subgraph-querying infrastructure. We substantiate the benefits of Lindex and its disk-resident variation Lindex+ theoretically and empirically.     

The possible cardinalities of global secure sets in cographs

Let G=(V,E) be a graph. A global secure set SD⊆V is a dominating set which also satisfies a condition that |N[X]∩SD|≥|N[X]−SD| for every subset X⊆SD. The minimum cardinality of the global secure set in the graph G is denoted by γ_s(G). In this paper, we introduce the notion of γ_s-monotone graphs. The graph G is γ_s-monotone if, for every k∈{γ_s(G),γ_s(G)+1,…,n}, it has a global secure set of cardinality k. We will also present the results concerning the minimum cardinality of the global secure sets in the class of cographs.     

RBF neural network based on q-Gaussian function in function approximation

To enhance the generalization performance of radial basis function (RBF) neural networks, an RBF neural network based on a q-Gaussian function is proposed. A q-Gaussian function is chosen as the radial basis function of the RBF neural network, and a particle swarm optimization algorithm is employed to select the parameters of the network. The non-extensive entropic index q is encoded in the particle and adjusted adaptively in the evolutionary process of population. Simulation results of the function approximation indicate that an RBF neural network based on q-Gaussian function achieves the best generalization performance.     

Approximate minimum weight matching on points ink-dimensional space

We study the problem of finding a minimum weight complete matching in the complete graph on a set V ofn points ink-dimensional space. The points are the vertices of the graph and the weight of an edge between any two points is the distance between the points under someL _q,-metric. We give anO((2c _q )^1.5k ?^?1.5k (α(n, n))^0.5 n ^1.5(logn)^2.5) algorithm for finding an almost minimum weight complete matching in such a graph, wherec _q =6k ^1/q for theL _q -metric, α is the inverse Ackermann function, and ? ≤ 1. The weight of the complete matching obtained by our algorithm is guaranteed to be at most (1 + ?) times the weight of a minimum weight complete matching.     

Wiser: A semantic approach for expert finding in academia based on entity linking

We present Wiser, a new semantic search engine for expert finding in academia. Our system is unsupervised and it jointly combines classical language modeling techniques, based on text evidences, with the Wikipedia Knowledge Graph, via entity linking.Wiser indexes each academic author through a novel profiling technique which models her expertise with a small, labeled and weighted graph drawn from Wikipedia. Nodes in this graph are the Wikipedia entities mentioned in the author’s publications, whereas the weighted edges express the semantic relatedness among these entities computed via textual and graph-based relatedness functions. Every node is also labeled with a relevance score which models the pertinence of the corresponding entity to author’s expertise, and is computed by means of a proper random-walk calculation over that graph; and with a latent vector representation which is learned via entity and other kinds of structural embeddings derived from Wikipedia.At query time, experts are retrieved by combining classic document-centric approaches, which exploit the occurrences of query terms in the author’s documents, with a novel set of profile-centric scoring strategies, which compute the semantic relatedness between the author’s expertise and the query topic via the above graph-based profiles.The effectiveness of our system is established over a large-scale experimental test on a standard dataset for this task. We show that Wiser achieves better performance than all the other competitors, thus proving the effectiveness of modeling author’s profile via our “semantic” graph of entities. Finally, we comment on the use of Wiser for indexing and profiling the whole research community within the University of Pisa, and its application to technology transfer in our University.     

Embeddings of Hypercubes and Grids into de Bruijn Graphs

An embedding of a graph G into a graph H is an injective mapping f from the vertices of G into the vertices of H together with a mapping P_f of edges of G into paths in H. The dilation of the embedding is tile maximum taken over all the lengths of the paths P_f(xy) associated with the edges xy of G. We show that it is possible to embed a D-dimensional hypercube into the binary de Bruijn graph of the same order and diameter with dilation at most [D/2]. Similarly a majority of planar grids can be embedded into a binary de Bruijn graph of the same or nearly the same order with dilation at most [D/2] where D is the diameter of the de Bruijn graph.     

Graph optimization for dimensionality reduction with sparsity constraints

Graph-based dimensionality reduction (DR) methods play an increasingly important role in many machine learning and pattern recognition applications. In this paper, we propose a novel graph-based learning scheme to conduct Graph Optimization for Dimensionality Reduction with Sparsity Constraints (GODRSC). Different from most of graph-based DR methods where graphs are generally constructed in advance, GODRSC aims to simultaneously seek a graph and a projection matrix preserving such a graph in one unified framework, resulting in an automatically updated graph. Moreover, by applying an l₁ regularizer, a sparse graph is achieved, which models the “locality” structure of data and contains natural discriminating information. Finally, extensive experiments on several publicly available UCI and face databases verify the feasibility and effectiveness of the proposed method.     

Efficient distributed approximation algorithms via probabilistic tree embeddings

We present a uniform approach to design efficient distributed approximation algorithms for various fundamental network optimization problems. Our approach is randomized and based on a probabilistic tree embedding due to Fakcharoenphol et?al. (J Comput Syst Sci 69(3):485–497, 2004) (FRT embedding). We show how to efficiently compute an (implicit) FRT embedding in a decentralized manner and how to use the embedding to obtain efficient expected O(log n)-approximate distributed algorithms for various problems, in particular the generalized Steiner forest problem (including the minimum Steiner tree problem), the minimum routing cost spanning tree problem, and the k-source shortest paths problem. The distributed construction of the FRT embedding is based on the computation of least elements (LE) lists, a distributed data structure that is of independent interest. Assuming a global order on the nodes of a network, the LE-list of a node stores the smallest node (w.r.t. the given order) within every distance d (cf. Cohen in J Comput Syst Sci 55(3):441–453, 1997, Cohen and Kaplan in J Comput Syst Sci 73(3):265–288, 2007). Assuming a random order on the nodes, we give a distributed algorithm for computing LE-lists on a weighted graph with time complexity O(S log n), where S is a graph parameter called the shortest path diameter which can be considered the weighted counterpart of the diameter D of the graph. For unweighted graphs, our LE-lists computation has asymptotically optimal time complexity of O(D). As a byproduct, we get an improved synchronous leader election algorithm for general networks that is both time-optimal and almost message-optimal with high probability.     

There are planar graphs almost as good as the complete graphs and almost as cheap as minimum spanning trees

Let S be a set ofn points in the plane. For an arbitrary positive rationalr, we construct a planar straight-line graph onS that approximates the complete Euclidean graph onS within the factor (1 + 1/r)[2π/3 cos(π/6)], and it has length bounded by 2r + 1 times the length of a minimum Euclidean spanning tree onS. Given the Deiaunay triangulation ofS, the graph can be constructed in linear time.     

Algorithms for derivation of structurally stable Hamiltonian signed graphs

A graph S_p,q,n refers to a signed graph with p nodes and q edges with n being the number of negative edges. We introduce two theorems to facilitate identification of the complete set of balanced signed graph configurations for any p-node Hamiltonian signed graph in terms of p, q and n. This allows for the development of computational procedures to efficiently determine the structural stability of a signed graph. This is potentially useful for the planning and analysis of complex situations or scenarios which can be depicted as signed graphs. Through the application of the theorems, the state of balance of a signed graph structure or its affinity towards balance can be determined in a more time-efficient manner compared to any explicit enumeration algorithm.     

Proof system representations of degrees of disjoint NP-pairs

Let D be a set of many-one degrees of disjoint NP-pairs. We define a proof system representation of D to be a set of propositional proof systems P such that each degree in D contains the canonical NP-pair of a corresponding proof system in P and the degree structure of D is reflected by the simulation order among the corresponding proof systems in P. We also define a nesting representation of D to be a set of NP-pairs S such that each degree in D contains a representative NP-pair in S and the degree structure of D is reflected by the inclusion relations among their representative NP-pairs in S. We show that proof system and nesting representations both exist for D if the lower span of each degree in D overlaps with D on a finite set only. In particular, a linear chain of many-one degrees of NP-pairs has both a proof system representation and a nesting representation. This extends a result by Glaßer et al. (2009). We also show that in general D has a proof system representation if it has a nesting representation where all representative NP-pairs share the same set as their first components.     

Inapproximability of the Tutte polynomial of a planar graph

The Tutte polynomial of a graph G is a two-variable polynomial T(G; x, y) that encodes many interesting properties of the graph. We study the complexity of the following problem, for rationals x and y: given as input a planar graph G, determine T(G; x, y). Vertigan completely mapped the complexity of exactly computing the Tutte polynomial of a planar graph. He showed that the problem can be solved in polynomial time if (x, y) is on the hyperbola H _q given by (x ? 1)(y ? 1) = q for q = 1 or q = 2 or if (x, y) is one of the two special points (x, y) = (?1, ?1) or (x, y) = (1, 1). Otherwise, the problem is #P-hard. In this paper, we consider the problem of approximating T(G; x, y), in the usual sense of “fully polynomial randomized approximation scheme” or FPRAS. Roughly speaking, an FPRAS is required to produce, in polynomial time and with high probability, an answer that has small relative error. Assuming that NP is different from RP, we show that there is no FPRAS for the Tutte polynomial in a large portion of the (x, y) plane. In particular, there is no FPRAS if x > 1, y < ?1 or if y > 1, x < ?1 or if x < 0, y < 0 and q > 5. Also, there is no FPRAS if x < 1, y < 1 and q = 3. For q > 5, our result is intriguing because it shows that there is no FPRAS at (x, y) =?(1 ? q/(1 + ε), ?ε) for any positive ε but it leaves open the limit point ε =?0, which corresponds to approximately counting q-colorings of a planar graph.     

Fault-diameter of Cartesian graph bundles

Cartesian graph bundles is a class of graphs that is a generalization of the Cartesian graph products. Let G be a kG-connected graph and Dc(G) denote the diameter of G after deleting any of its c<kG vertices. We prove that Da+b+1(G)?Da(F)+Db(B)+1 if G is a graph bundle with fibre F over base B, a<kF, and b<kB.     

Using stable sets to bound the chromatic number

A generalization of the Roy-Gallai theorem is presented: it is based on the existence in any oriented graph of a stable set S such that for any node w not in S there is an elementary path from some node of S to w. The bound obtained improves earlier results by Berge and by Li.     

A simple proof of Rosenfeld's characterization of digital straight line segments

A digital straight line segment is defined as the grid-intersect quantization of a straight line segment in the plane. Let S be a set of pixels on a square grid. Rosenfeld [8] showed that S is a digital straight line segment if and only if it is a Digital arc having the chord property. Then Kim and Rosenfeld [3,6] showed that S has the chord properly if and if for every p, q?S there is a digital straight line segment C ? S such that p and q are the extremities of C.We give a simple proof of these two results based on the Transversal Theorem of Santaló. We show how the underlying methodology can be generalized to the case of (infinite) digital straight lines and to the quantization of hyperplanes in an n-dimensional space for n ≥ 3.     

Computing maximal Kleene closures that are embeddable in a given subword-closed language

Motivated by certain coding techniques for reliable DNA computing, we consider the problem of characterizing nontrivial languages D that are maximal with the property that D ^* is contained in the subword closure of a given set S of words of some fixed length k. This closure is simply the set of all words whose subwords of length k must be in S. We provide a deep structural characterization of these languages D, which leads to polynomial time algorithms for computing such languages.     

The Outer-connected Domination Number of Sierpiński-like Graphs

An outer-connected dominating set in a graph G = (V, E) is a set of vertices D ? V satisfying the condition that, for each vertex v ? D, vertex v is adjacent to some vertex in D and the subgraph induced by V?D is connected. The outer-connected dominating set problem is to find an outer-connected dominating set with the minimum number of vertices which is denoted by \(\tilde {\gamma }_{c}(G)\). In this paper, we determine \(\tilde {\gamma }_{c}(S(n,k))\), \(\tilde {\gamma }_{c}(S^{+}(n,k))\), \(\tilde {\gamma }_{c}(S^{++}(n,k))\), and \(\tilde {\gamma }_{c}(S_{n})\), where S(n, k), S ⁺(n, k), S ⁺⁺(n, k), and S _n are Sierpi\(\acute {\mathrm {n}}\)ski-like graphs.     

Two algorithms for minimum 2-connected r-hop dominating set

For a graph G=(V,E), a subset D⊆V is an r-hop dominating set if every vertex u∈V−D is at most r-hops away from D. It is a 2-connected r-hop dominating set if the subgraph of G induced by D is 2-connected. In this paper, we present two approximation algorithms to compute minimum 2-connected r-hop dominating set. The first one is a greedy algorithm using ear decomposition of 2-connected graphs. This algorithm is applicable to any 2-connected general graph. The second one is a three-phase algorithm which is only applicable to unit disk graphs. For both algorithms, performance ratios are given.     

Computer package for microscopic spin Hamiltonian analysis of the 3d4 and 3d6 (spin S = 2) ions at orthorhombic and tetragonal symmetry sites

A FORTRAN package based on the microscopic spin Hamiltonian expressions derived by computer algebra (ALTRAN) for the high spin (S = 2) 3d⁴ and 3d⁶ ions with an orbital singlet ground state at orthorhombic (point groups: C_2v, D₂, D_2h) and tetragonal (point groups: C_4v, D₄, D_4h, D₂d) symmetry sites is presented. The spin—orbit (λ) and the spin—spin (ρ) coupling contributions up to the fourth-order perturbation theory are taken into account within the ⁵D approximation. The package enables efficient numerical calculations of the Zeeman electronic (Ze) parameters and the zero-field splitting (ZFS) parameters for the S = 2 3d⁴ and 3d⁶ ions. The following terms are included: λ and λ² for the Ze parameters, λ², λ³ λ⁴, ρ, ρ² and λ²ρ for the second-rank ZFS ones, and λ⁴, ρ² and λ²ρ for the fourth-rank ZFS ones. The program is applicable to all possible energy level schemes with a ground orbital singlet arising from the ⁵D multiplet due to orthorhombic or tetragonal symmetry crystal fields. The input parameters are λ, ρ, the energy levels Δ_j (j = 1, 2, 3, 4) and the mixing coefficient s, which can be obtained from other spectroscopic data. The ZFS parameters output in the extended Stevens notation B_k^q and b_k^q, as well as the conventional notation (D, E; a, F, K), is provided.