We commence the study of domination in the incidence graphs of combinatorial designs. Let @math be a combinatorial design and denote by @math the domination number of the incidence (Levy) graph of @math . We obtain a number of results about the domination numbers of various kinds of designs. For instance, a finite projective plane of order @math , which is a symmetric @math -design, has @math . We also show that for any symmetric @math -design it holds that @math . We study at depth the domination numbers of Steiner systems and in particular of Steiner triple systems. We show that a @math has @math and also obtain a number of upper bounds. The tantalizing conjecture that all Steiner triple systems on @math vertices have the same domination number is proposed and is verified up to @math . The structure of minimal dominating sets is also investigated, both for its own sake and as a tool in deriving lower bounds on @math . Finally, a number of open questions are proposed.
Laskar and Wallis in @cite_8 have obtained some results about the domination number of the line graph of @math , motivated by combinatorial chessboard problems. However, the domination number of @math itself was not considered by them. The recent thesis of H ' e ger @cite_4 is probably closeset in spirit to the present work: H ' e ger constructs dominating sets of a particular form in projective planes.
@cite_4 @cite_8
2064550979 1995808019
The graph-theoretic study of combinatorial chessboard problems can be extended to the study of line graphs of graphs of combinatorial designs. In particular, the determination of optimal placements of rooks on a chessboard corresponds to the determination of domination parameters of graphs of block designs. The determination of one such parameter, the independence number, is shown to follow directly from classical results in design theory. Additionally, the domination number of graphs of finite projective planes is discussed. Introduction (S.T. Hedetniemi, R.C. Laskar). Theoretical. Chessboard Domination Problems (E.J. Cockayne). On the Queen Domination Problem (C.M. Grinstead, B. Hahne, D. Van Stone). Recent Problems and Results About Kernels in Directed Graphs (C. Berge, P. Duchet). Critical Concepts in Domination (D.P. Summer). The Bondage Number of a Graph (J.F. ). Chordal Graphs and Upper Irredundance, Upper Domination and Independence (M.S. Jacobson and K. Peters). Regular Totally Domatically Full Graphs (B. Zelinka). Domatically Critical and Domatically Full Graphs (D. Rall). On Generalised Minimal Domination Parameters for Paths (B. Bollobas, E.J. Cockayne, C.M. Mynhardt). New Models. Dominating Cliques in Graphs (M.B. Cozzens, L.L. Kelleher). Covering all Cliques of Graph (Z. Tuza). Factor Domination in Graphs (R.C. Brigham, R. Dutton). The Least Point Covering and Domination Numbers of a Graph (E. Sampathkumar). Algorithmic. Dominating Sets in Perfect Graphs (D.G. Corneil, L.K. Stewart). Unit Disk Graphs (B.N. Clark, C.J. Colbourn, D.S. Johnson). Permutation Graphs: Connected Domination and Steiner Trees (C.J. Colbourn, L.K. Stewart). The Discipline Number of a Graph (V. Chvatal, W.J. Cook). Best Location of Service Centers in a Tree-Like Network under Budget Constraints (J. McHugh, Y. Perl). Dominating Cycles in Halin Graphs (M. Skowronska, M.M. Syslo). Finding Dominating Cliques Efficiently, in Strongly Chordal Graphs and Undirected Path Graphs (D. Kratsch). On Minimum Dominating Sets with Minimum Intersection (D.L. Grinstead, P.J. Slater). Bibliography. Bibliography on Domination in Graphs and Some Basic Definitions of Domination Parameters (S.T. Hedetniemi, R.C. Laskar). Let @math be a 3-uniform hypergraph of order @math and size @math , and let @math be a subset of vertices of @math . The set @math is a strong transversal in @math if @math contains at least two vertices from every edge of @math . The strong transversal number @math of @math is the minimum size of a strong transversal in @math . We show that @math , and we characterize the hypergraphs that achieve equality in this bound. In particular, we show that the Fano plane is the only connected 3-uniform hypergraph @math of order @math and size @math that achieves equality in this bound. A set @math of vertices in a graph @math is a double total dominating set of @math if every vertex of @math is adjacent to at least two vertices in @math . The minimum cardinality of a double total dominating set of @math is the double total domination number @math of @math . Let @math be a connected graph of order @math with minimum degree at least three. As an application of our hypergraph results, we show that @math with equality if and only if @math is the Heawood graph (equivalently, the incidence bipartite graph of the Fano plane). Further if @math is not the Heawood graph, we show that @math , while if @math is a cubic graph different from the Heawood graph, we show that @math , and this bound is sharp. A set @math is weakly convex in @math if for any two vertices @math there exists an @math --geodesic such that all of its vertices belong to @math . A set @math is a weakly convex dominating set if @math is weakly convex and dominating. The weakly convex domination number @math of a graph @math equals the minimum cardinality of a weakly convex dominating set in @math . The weakly convex domination subdivision number sd @math is the minimum number of edges that must be subdivided (each edge in @math can be subdivided at most once) in order to increase the weakly convex domination number. In this paper we initiate the study of weakly convex domination subdivision number and we establish upper bounds for it. All graphs considered in this paper are finite graphs without loops and multiple edges. The domatic number of a graph was defined by E. J. Cockayne and S. T. Hedetniemi [1]. Later some related concepts were introduced. The same authors together with R. M. Dawes [2] have introduced the total domatic number; R. Laskar and S. T. Hedetniemi [3] have introduced the connected domatic number. A dominating set (or a total dominating set) in an undirected graph G is a subset D of the vertex set V(G) of G with the property that to each vertex x e V(G) — D (or to each vertex xeV(G) respectively) there exists a vertex y e D adjacent to x. A connected dominating set of G is a dominating set of G with the property that the subgraph of G induced by it is connected. A domatic (or total domatic, or connected domatic) partition of G is a partition of V(G), all of whose classes are dominating (or total dominating, or connected dominating, respectively) sets of G. The maximum number of classes of a domatic (or total domatic, or connected domatic) partition of G is called the domatic (or total domatic, or connected domatic, respectively) number of G. The domatic number of G is denoted by d(G), its total domatic number by <4(G), its connected domatic number by dc(G). The connected domatic number of a graph is well defined only for connected graphs; in a disconnected graph there exists no connected dominating set and thus no connected domatic partition, while in every connected graph there exists at least one connected domatic partition, namely that which consists of one class. The connected domatic number of G is closely related to the vertex connectivity number of G. If G is a connected graph, then a vertex cut of G is a subset R of V(G) with the property that the subgraph of G induced by V(G) — R is disconnected. If G is not a complete graph, then the vertex connectivity number x(G) is the minimum cardinality of a vertex cut of G. If G is a complete graph (i. e. without vertex cuts) with n vertices, then we put x(G) = n — 1. Lemma. Let G be a connected graph which is not complete, let R be its vertex cut, let D be its connected dominating set. Then DnR^0. A set of vertices in a graph is a dominating set if every vertex outside the set has a neighbor in the set. The domatic number problem is that of partitioning the vertices of a graph into the maximum number of disjoint dominating sets. Let n denote the number of vertices, @math the minimum degree, and @math the maximum degree. We show that every graph has a domatic partition with @math dominating sets and, moreover, that such a domatic partition can be found in polynomial-time. This implies a @math -approximation algorithm for domatic number, since the domatic number is always at most @math . We also show this to be essentially best possible. Namely, extending the approximation hardness of set cover by combining multiprover protocols with zero-knowledge techniques, we show that for every @math , a @math -approximation implies that @math . This makes domatic number the first natural maximization problem (known to the authors) that is provably approximable to within polylogarithmic factors but no better. We also show that every graph has a domatic partition with @math dominating sets, where the "o(1)" term goes to zero as @math increases. This can be turned into an efficient algorithm that produces a domatic partition of @math sets. We consider the problem of fitting the parameters of a high-dimensional linear regression model. In the regime where the number of parameters @math is comparable to or exceeds the sample size @math , a successful approach uses an @math -penalized least squares estimator, known as Lasso. Unfortunately, unlike for linear estimators (e.g., ordinary least squares), no well-established method exists to compute confidence intervals or p-values on the basis of the Lasso estimator. Very recently, a line of work javanmard2013hypothesis, confidenceJM, GBR-hypothesis has addressed this problem by constructing a debiased version of the Lasso estimator. In this paper, we study this approach for random design model, under the assumption that a good estimator exists for the precision matrix of the design. Our analysis improves over the state of the art in that it establishes nearly optimal testing power if the sample size @math asymptotically dominates @math , with @math being the sparsity level (number of non-zero coefficients). Earlier work obtains provable guarantees only for much larger sample size, namely it requires @math to asymptotically dominate @math . In particular, for random designs with a sparse precision matrix we show that an estimator thereof having the required properties can be computed efficiently. Finally, we evaluate this approach on synthetic data and compare it with earlier proposals. We consider multiobjective and parametric versions of the global minimum cut problem in undirected graphs and bounded-rank hypergraphs with multiple edge cost functions. For a fixed number of edge cost functions, we show that the total number of supported non-dominated (SND) cuts is bounded by a polynomial in the numbers of nodes and edges, i.e., is strongly polynomial. This bound also applies to the combinatorial facet complexity of the problem, i.e., the maximum number of facets (linear pieces) of the parametric curve for the parametrized (linear combination) objective, over the set of all parameter vectors such that the parametrized edge costs are nonnegative and the parametrized cut costs are positive. We sharpen this bound in the case of two objectives (the bicriteria problem), for which we also derive a strongly polynomial upper bound on the total number of non-dominated (Pareto optimal) cuts. In particular, the bicriteria global minimum cut problem in an n-node graph admits @math O(n3logn) SND cuts and @math O(n5logn) Pareto optimal cuts. These results significantly improve on earlier graph cut results by Mulmuley (SIAM J Comput 28(4):1460---1509, 1999) and Armon and Zwick (Algorithmica 46(1):15---26, 2006). They also imply that the parametric curve and all SND cuts, and, for the bicriteria problems, all Pareto optimal cuts, can be computed in strongly polynomial time when the number of objectives is fixed. In this paper, we study the dominance relation under a stochastic setting. Let @math be a set of @math colored stochastic points in @math , each of which is associated with an existence probability. We investigate the problem of computing the probability that a realization of @math contains inter-color dominances, which we call the (CSD) problem. We propose the first algorithm to solve the CSD problem for @math in @math time. On the other hand, we prove that, for @math , even the CSD problem with a restricted color pattern is #P-hard. In addition, even if the existence probabilities are restricted to be @math , the problem remains #P-hard for @math . A simple FPRAS is then provided to approximate the desired probability in any dimension. We also study a variant of the CSD problem in which the dominance relation is considered with respect to not only the standard basis but any orthogonal basis of @math . Specifically, this variant, which we call the free-basis colored stochastic dominance (FBCSD) problem, considers the probability that a realization of @math contains inter-color dominances with respect to any orthogonal basis of @math . We show that the CSD problem is polynomial-time reducible to the FBCSD problem in the same dimension, which proves the #P-hardness of the latter for @math . Conversely, we reduce the FBCSD problem in @math to the CSD problem in @math , by which an @math time algorithm for the former is obtained. Problems of finding p-centers and dominating sets of radius r in networks are discussed in this paper. Let n be the number of vertices and @math be the number of edges of a network. With the assumption that the distance-matrix of the network is available, we design an @math algorithm for finding an absolute 1-center of a vertex-weighted network and an @math algorithm for finding an absolute 1-center of a vertex-unweighted network (the problem of finding a vertex 1-center of a network is trivial). We show that the problem of finding a (vertex or absolute) p-center (for @math ) of a (vertex-weighted or vertex-unweighted) network, and the problem of finding a dominating set of radius r are @math -hard even in the case where the network has a simple structure (e.g., a planar graph of maximum vertex degree 3). However, we describe an algorithm of complexity @math (respectively, $ O[| E |^p n^ 2p - 1... We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math . A dominating set in a graph G is a connected dominating set of G if it induces a connected subgraph of G. The connected domatic number of G is the maximum number of pairwise disjoint, connected dominating sets in V(G). We establish a sharp lower bound on the number of edges in a connected graph with a given order and given connected domatic number. We also show that a planar graph has connected domatic number at most 4 and give a characterization of planar graphs having connected domatic number 3. Millions of images on the web enable us to explore images from social events such as a family party, thus it is of interest to understand and model the affect exhibited by a group of people in images. But analysis of the affect expressed by multiple people is challenging due to varied indoor and outdoor settings, and interactions taking place between various numbers of people. A few existing works on Group-level Emotion Recognition (GER) have investigated on face-level information. Due to the challenging environments, face may not provide enough information to GER. Relatively few studies have investigated multi-modal GER. Therefore, we propose a novel multi-modal approach based on a new feature description for understanding emotional state of a group of people in an image. In this paper, we firstly exploit three kinds of rich information containing face, upperbody and scene in a group-level image. Furthermore, in order to integrate multiple person's information in a group-level image, we propose an information aggregation method to generate three features for face, upperbody and scene, respectively. We fuse face, upperbody and scene information for robustness of GER against the challenging environments. Intensive experiments are performed on two challenging group-level emotion databases to investigate the role of face, upperbody and scene as well as multi-modal framework. Experimental results demonstrate that our framework achieves very promising performance for GER. A set L@?V(G) of a graph G=(V,E) is a liar's dominating set if (1) for all v@?V(G), |N"G[v]@?L|>=2 and (2) for every pair u,v@?V(G) of distinct vertices, |(N"G[u]@?N"G[v])@?L|>=3. A graph G=(V,E) admits a liar's dominating set if each of its connected component has at least three vertices. Given a graph G=(V,E) and an integer K, the liar's domination decision problem (LR-DOMDP) is to decide whether G has a liar's dominating set of cardinality at most K. Slater [P.J. Slater, Liar's Domination, Networks, 54(2) (2009) 70-74] proved that the LR-DOMDP is NP-complete for general graphs. Subsequently, Roden and Slater [M.L. Roden and P.J. Slater, Liar's domination in graphs, Discrete Math., 309(19) (2009) 5884-5890] showed a more general family of problems to each be NP-complete for bipartite graphs. Besides this result, no other algorithmic result for the liar's dominating set problem is available in the literature. In this paper, we first strengthen the complexity result of the LR-DOMDP by showing that this problem remains NP-complete for split graphs and hence for chordal graphs. Finally, we propose a linear time algorithm for computing a minimum cardinality liar's dominating set in a tree. We show that every weighted connected graph @math contains as a subgraph a spanning tree into which the edges of @math can be embedded with average stretch @math . Moreover, we show that this tree can be constructed in time @math in general, and in time @math if the input graph is unweighted. The main ingredient in our construction is a novel graph decomposition technique. Our new algorithm can be immediately used to improve the running time of the recent solver for symmetric diagonally dominant linear systems of Spielman and Teng from @math to @math , and to @math when the system is planar. Our result can also be used to improve several earlier approximation algorithms that use low-stretch spanning trees. We prove an incidence theorem for points and planes in the projective space @math over any field @math , whose characteristic @math An incidence is viewed as an intersection along a line of a pair of two-planes from two canonical rulings of the Klein quadric. The Klein quadric can be traversed by a generic hyperplane, yielding a line-line incidence problem in a three-quadric, the Klein image of a regular line complex. This hyperplane can be chosen so that at most two lines meet. Hence, one can apply an algebraic theorem of Guth and Katz, with a constraint involving @math if @math . This yields a bound on the number of incidences between @math points and @math planes in @math , with @math as @math where @math is the maximum number of collinear planes, provided that @math if @math . Examples show that this bound cannot be improved without additional assumptions. This gives one a vehicle to establish geometric incidence estimates when @math . For a non-collinear point set @math and a non-degenerate symmetric or skew-symmetric bilinear form @math , the number of distinct values of @math on pairs of points of @math is @math . This is also the best known bound over @math , where it follows from the Szemer 'edi-Trotter theorem. Also, a set @math , not supported in a single semi-isotropic plane contains a point, from which @math distinct distances to other points of @math are attained. The visibility graph of a set of nonintersecting polygonal obstacles in the plane is an undirected graph whose vertex set consists of the vertices of the obstacles and whose edges are pairs of vertices @math such that the open line segment between u and v does not intersect any of the obstacles. The visibility graph is an important combinatorial structure in computational geometry and is used in applications such as solving visibility problems and computing shortest paths. This paper presents an algorithm that computes the visibility graph of a set of obstacles in time @math , where E is the number of edges in the visibility graph and n is the total number of vertices in all the obstacles. We say a vertex v in a graph Gcovers a vertex w if v=w or if v and w are adjacent. A subset of vertices of G is a dominating set if it collectively covers all vertices in the graph. The dominating set problem, which is NP-hard, consists of finding a smallest possible dominating set for a graph. The straightforward greedy strategy for finding a small dominating set in a graph consists of successively choosing vertices which cover the largest possible number of previously uncovered vertices. Several variations on this greedy heuristic are described and the results of extensive testing of these variations is presented. A more sophisticated procedure for choosing vertices, which takes into account the number of ways in which an uncovered vertex may be covered, appears to be the most successful of the algorithms which are analyzed. For our experimental testing, we used both random graphs and graphs constructed by test case generators which produce graphs with a given density and a specified size for the smallest dominating set. We found that these generators were able to produce challenging graphs for the algorithms, thus helping to discriminate among them, and allowing a greater variety of graphs to be used in the experiments. Unit disk graphs are the intersection graphs of equal sized circles in the plane: they provide a graph-theoretic model for broadcast networks (cellular networks) and for some problems in computational geometry. We show that many standard graph theoretic problems remain NP-complete on unit disk graphs, including coloring, independent set, domination, independent domination, and connected domination; NP-completeness for the domination problem is shown to hold even for grid graphs, a subclass of unit disk graphs. In contrast, we give a polynomial time algorithm for finding cliques when the geometric representation (circles in the plane) is provided. We initiate a study of the homomorphism domination exponent of a pair of graphs F and G, defined as the maximum real number c such that |Hom(F,T)|>=|Hom(G,T)|^c for every graph T. The problem of determining whether HDE(F,G)>=1 is known as the homomorphism domination problem, and its decidability is an important open question arising in the theory of relational databases. We investigate the combinatorial and computational properties of the homomorphism domination exponent, proving upper and lower bounds and isolating classes of graphs F and G for which HDE(F,G) is computable. In particular, we present a linear program computing HDE(F,G) in the special case, where F is chordal and G is series-parallel.