Showing papers on "Quartic graph published in 2004"
01 Jan 2004
TL;DR: This work gives a precise characterization of what energy functions can be minimized using graph cuts, among the energy functions that can be written as a sum of terms containing three or fewer binary variables.
Abstract: In the last few years, several new algorithms based on graph cuts have been developed to solve energy minimization problems in computer vision. Each of these techniques constructs a graph such that the minimum cut on the graph also minimizes the energy. Yet, because these graph constructions are complex and highly specific to a particular energy function, graph cuts have seen limited application to date. In this paper, we give a characterization of the energy functions that can be minimized by graph cuts. Our results are restricted to functions of binary variables. However, our work generalizes many previous constructions and is easily applicable to vision problems that involve large numbers of labels, such as stereo, motion, image restoration, and scene reconstruction. We give a precise characterization of what energy functions can be minimized using graph cuts, among the energy functions that can be written as a sum of terms containing three or fewer binary variables. We also provide a general-purpose construction to minimize such an energy function. Finally, we give a necessary condition for any energy function of binary variables to be minimized by graph cuts. Researchers who are considering the use of graph cuts to optimize a particular energy function can use our results to determine if this is possible and then follow our construction to create the appropriate graph. A software implementation is freely available.
3,079 citations
TL;DR: The tool described here is the first part of a tool set called GROOVE (GRaph-based Object-Oriented VErification) for software model checking of object-oriented systems using graphs to represent state snapshots; transitions arise from the application of graph production rules.
Abstract: The tool described here is the first part of a tool set called GROOVE (GRaph-based Object-Oriented VErification) for software model checking of object-oriented systems. The special feature of GROOVE, which sets it apart from other model checking approaches, is that it is based on graph transformations. It uses graphs to represent state snapshots; transitions arise from the application of graph production rules. This yields so-called Graph Transition Systems (GTSrsquos) as computational models.
352 citations
TL;DR: In this article, the authors provide transformation rules, stated in purely graph theoretical terms, which completely characterize the evolution of graph states under local Clifford operations, i.e., successive application of which generates the orbit of any graph state under local unitary operations within the Clifford group.
Abstract: We translate the action of local Clifford operations on graph states into transformations on their associated graphs, i.e., we provide transformation rules, stated in purely graph theoretical terms, which completely characterize the evolution of graph states under local Clifford operations. As we will show, there is essentially one basic rule, successive application of which generates the orbit of any graph state under local unitary operations within the Clifford group.
267 citations
TL;DR: It is proved that A/sub n,k/-F is Hamiltonian if |F|/spl les/k(n-k)-2 and A/ sub n, k/-F are Hamiltonian connected if |f|/ spl les/ k(n -k)-3.
Abstract: The arrangement graph A/sub n,k/ is a generalization of the star graph. There are some results concerning fault Hamiltonicity and fault Hamiltonian connectivity of the arrangement graph. However, these results are restricted in some particular cases and, thus, are less completed. We improve these results and obtain a stronger and simpler statement. Let n-k/spl ges/2 and F/spl sube/V(A/sub n,k/)/spl cup/E(A/sub n,k/). We prove that A/sub n,k/-F is Hamiltonian if |F|/spl les/k(n-k)-2 and A/sub n,k/-F is Hamiltonian connected if |F|/spl les/k(n-k)-3. These results are optimal.
99 citations
21 Jun 2004
TL;DR: In this paper, a preprocessing technique based on crown decompositions of an auxiliary graph is proposed to reduce an arbitrary input graph of the problem to a graph on O(k 3) vertices in polynomial time.
Abstract: We consider the NP-complete problem of deciding whether an input graph on n vertices has k vertex-disjoint copies of a fixed graph H. For H=K3 (the triangle) we give an O(22klog k+1.869kn2) algorithm, and for general H an O(2k|H|logk+2k|H|log |H|n|H|) algorithm. We introduce a preprocessing (kernelization) technique based on crown decompositions of an auxiliary graph. For H=K3 this leads to a preprocessing algorithm that reduces an arbitrary input graph of the problem to a graph on O(k3) vertices in polynomial time.
71 citations
Journal Article•
TL;DR: It is shown that, for any K-state robot and any d > 3, there exists a planar graph of maximum degree d with at most K + 1 nodes that the robot cannot explore, and proved that the worst case space complexity of graph exploration is Θ(D log d) bits.
Abstract: A finite automaton, simply referred to as a robot, has to explore a graph whose nodes are unlabeled and whose edge ports are locally labeled at each node. The robot has no a priori knowledge of the topology of the graph or of its size. Its task is to traverse all the edges of the graph. We first show that, for any K-state robot and any d > 3, there exists a planar graph of maximum degree d with at most K + 1 nodes that the robot cannot explore. This bound improves all previous bounds in the literature. More interestingly, we show that, in order to explore all graphs of diameter D and maximum degree d, a robot needs Ω(D log d) memory bits, even if we restrict the exploration to planar graphs. This latter bound is tight. Indeed, a simple DFS at depth D + 1 enables a robot to explore any graph of diameter D and maximum degree d using a memory of size O(D log d) bits. We thus prove that the worst case space complexity of graph exploration is Θ(D log d) bits.
61 citations
TL;DR: It is proved that there exists a cyclic Hamiltonian k-cycle system of the complete graph if and only if k is odd but k≠15 and pα with p prime and α>1.
60 citations
TL;DR: The zigzag (or central circuit) structure of the resulting graph is studied using the algebraic formalism of the moving group, the $(k,l)-product and a finite index subgroup of $SL_2(\Bbb{Z})$, whose elements preserve the above structure.
Abstract: We consider the Goldberg-Coxeter construction $GC_{k,l}(G_0)$ (a generalization of a simplicial subdivision of the dodecahedron considered by Goldberg [Tohoku Mathematical Journal, 43 (1937) 104–108] and Coxeter [A Spectrum of Mathematics, OUP, (1971) 98–107]), which produces a plane graph from any $3$- or $4$-valent plane graph for integer parameters $k,l$. A zigzag in a plane graph is a circuit of edges, such that any two, but no three, consecutive edges belong to the same face; a central circuit in a $4$-valent plane graph $G$ is a circuit of edges, such that no two consecutive edges belong to the same face. We study the zigzag (or central circuit) structure of the resulting graph using the algebraic formalism of the moving group , the $(k,l)$-product and a finite index subgroup of $SL_2(\Bbb{Z})$, whose elements preserve the above structure. We also study the intersection pattern of zigzags (or central circuits) of $GC_{k,l}(G_0)$ and consider its projections , obtained by removing all but one zigzags (or central circuits).
47 citations
TL;DR: Efficient methods are developed for decomposing the graphs into subgraphs and healing the sub graphs to maintain the information corresponding to the eigenvalues and eigenvectors of the original graph for calculating the natural frequency of symmetric structures.
42 citations
11 Oct 2004
TL;DR: Experimental results show the superiority of the ACG (adjacent constraint graph) representation as a general floorplan representation, which has advantages of both adjacency graph and constraint graph of a floorplan.
Abstract: ACG (adjacent constraint graph) is invented as a general floorplan representation. It has advantages of both adjacency graph and constraint graph of a floorplan: edges in an ACG are between modules close to each other, thus the physical distance of two modules can be measured directly in the graph; since an ACG is a constraint graph, the floorplan area and module positions can be simply found by longest path computations. A natural combination of horizontal and vertical relations within one graph renders a beautiful data structure with full symmetry. The direct correspondence between geometrical positions of modules and ACG structures also makes it easy to incrementally change a floorplan and evaluate the result. Experimental results show the superiority of this representation.
42 citations
TL;DR: A reduction method to determine the hamiltonian index of a graph $G$ with h(G)\geq 2$ is given here, which improves some known results of P.A. Catlin et al. and H.-J.
TL;DR: This paper shows that the incremental bond graph can serve also as a starting point for setting up symbolically the canonical form as well as the standard interconnection form of state equations used for robustness study.
Journal Article•
TL;DR: In this article, the asymptotic behaviour of the neighborhood of the graph of a type of rapidly oscillating continuous functions is studied and necessary and sucient conditions for rapid oscillations of solutions of the main equation are given.
Abstract: We study the asymptotic behaviour of "-neighbourhood of the graph of a type of rapidly oscillating continuous functions. Next, we estate necessary and sucient conditions for rapid oscillations of solutions of the main equation. This enables us to verify some new singular properties of bounded continuous solutions of a class of nonlinear p-Laplacian by calculating lower and upper bounds for the Minkowski content and the s-dimensional density of the graph of each solution and its derivative.
02 Jun 2004
TL;DR: A new definition of an implicit surface over a noisy point cloud is presented that can be evaluated very fast, but, unlike other definitions based on the moving least squares approach, it does not suffer from artifacts.
Abstract: We present a new definition of an implicit surface over a noisy point cloud. It can be evaluated very fast, but, unlike other definitions based on the moving least squares approach, it does not suffer from artifacts.
In order to achieve robustness, we propose to use a different kernel function that approximates geodesic distances on the surface by utilizing a geometric proximity graph. The starting point in the graph is determined by approximate nearest neighbor search. From a variety of possibilities, we have examined the Delaunay graph and the sphere-of-influence graph (SIG). For both, we propose to use modifications, the r-SIG and the pruned Delaunay graph.
We have implemented our new surface definition as well as a test environment which allows to visualize and to evaluate the quality of the surfaces. We have evaluated the different surfaces induced by different proximity graphs. The results show that artifacts and the root mean square error are significantly reduced.
Journal Article•
TL;DR: It is shown that in any graph G on n vertices with d(x) + d(y) ≥ n for any two nonadjacent vertices x and y, the authors can fix the order of k vertices on a given cycle and find a hamiltonian cycle encountering these vertices in the same order, as long as k < n/12 and G is d(k + 1)/2e-connected.
Abstract: We show that in any graph G on n vertices with d(x) + d(y) ≥ n for any two nonadjacent vertices x and y, we can fix the order of k vertices on a given cycle and find a hamiltonian cycle encountering these vertices in the same order, as long as k < n/12 and G is d(k + 1)/2e-connected. Further we show that every b3k/2cconnected graph on n vertices with d(x) + d(y) ≥ n for any two nonadjacent vertices x and y is k-ordered hamiltonian, i.e. for every ordered set of k vertices we can find a hamiltonian cycle encountering these vertices in the given order. Both connectivity bounds are best possible.
TL;DR: In this paper, it was shown that common-divisor graphs for solvable groups have diameters of at most 3 and for nonsolvable groups their diameters are bounded above by 3.
Journal Article•
TL;DR: In this communication, degree splitting graph of a graph is defined and some properties ofdegree splitting graph are studied.
Abstract: In this communication we define degree splitting graph of a graph and we study some properties of degree splitting graph.
Journal Article•
TL;DR: A preprocessing technique based on crown decompositions of an auxiliary graph is introduced that leads to a preprocessing algorithm that reduces an arbitrary input graph of the problem to a graph on O(k3) vertices in polynomial time.
Abstract: We consider the NP-complete problem of deciding whether an input graph on n vertices has k vertex-disjoint copies of a fixed graph H. For H = K 3 (the triangle) we give an O(2 2k log k+1.869k n 2 ) algorithm, and for general H an O(2 k|H| log k+2k|H| log |H| n |H| ) algorithm. We introduce a preprocessing (kernelization) technique based on crown decompositions of an auxiliary graph. For H = K 3 this leads to a preprocessing algorithm that reduces an arbitrary input graph of the problem to a graph on O(k 3 ) vertices in polynomial time.
TL;DR: In this paper, the authors find families of prime diagrams of knots with arbitrary extreme coefficients in their Jones polynomials and give a positive answer to a question in their paper.
Abstract: We find families of prime diagrams of knots with arbitrary extreme coefficients in their Jones polynomials. Some graph theory is presents in connection with this problem, generalizing ideas by Yongju Bae and Morton [4] and giving a positive answer to a question in their paper.
TL;DR: A combinatorial method is developed to show that the dodecahedron graph has, up to rotation and reflection, a unique Hamiltonian cycle.
Abstract: We develop a combinatorial method to show that the dodecahedron graph has, up to rotation and reflection, a unique Hamiltonian cycle. Platonic graphs with this property are called topologically uniquely Hamiltonian. The same method is used to demonstrate topologically distinct Hamiltonian cycles on the icosahedron graph and to show that a regular graph embeddable on the 2-holed torus is topologically uniquely Hamiltonian.
Posted Content•
TL;DR: In this article, the amalgamated W probability space over the diagonal subalgebra DG, (W (G), E), where E : W (G) → DG is the conditional expectation is defined.
Abstract: In this paper, we will consider the Graph W -Probability Thoery. Let G be a countable directed graph. Then we have the the graph W -algebra defined by W (G) = span{Lw, L∗w′ : w, w ′ ∈ F+(G)} w , where F(G) is the free semigroupoid of G. Then naturally, we can define the amalgamated W probability space over the diagonal subalgebra DG, (W (G), E) , where E : W (G) → DG is the conditional expectation. We will consider the properties of this amalgamated W -probability space. As examples, we will observe the generating operator of W (G), and DG-even elements in (W (G), E) . In [8], Kribs and Power defined free semigroupoid algebras and they observed the properties of free semigroupoid algebras. In [14], on free semigroupoid algebras, we considered the probability theory by defining the conditional expectation E onto the diagonal subalgebra of the free semigroupoid algebra. Let G be a countable directed graph and let F(G) be the free semigroupoid in the sense of Kribs and Power. i.e F(G) is the collection of all vertices as units and all finite admissable paths on G. Let w be a finite path on G with its initial vertex x and its final vertex y. Then denote w by xwy, as usual. If HG is the generalized Fock space induced by the given graph, then we can define projections and partial isometries in B(HG), with respect to the vertices of G and finite paths on G, respectively. We will call the von Neumann algebra generated by such projections and partial isometris the graph W -algebra and denote it by W (G). We can define the diagonal subalgebra, DG, of W (G), which is unitarily isomorphic to ∆|G| ≃ C, where ∆|G| is the algebra generated by all diagonal matrices in the matricial algebra, M|G|(C). We say that the algebraic pair (W (G), E) is the graph W -probability space (over the diagonal subalgebra DG), where E : W (G) → DG is the suitable conditional expectation. We will consider the amalgamated free probability on this structure. It follows from the Speicher’s combinatorial amalgamated free probability knowledge. The von Neumann algebra W (G) is ; W (G) = C{Lw, Lw : w ∈ F+(G)} w , where Lw = L ∗ w is a projection if w is a vertex of G and Lw, L ∗ w are partial isometries if w is a finite (admissable) paths in F(G). If a ∈ (W (G), E) is a DG-valued random variable, then it has the following Fourier-like expression ;
Proceedings Article•
01 Jan 2004
TL;DR: In this paper, the design of boundary control laws for stabilizing systems of 2 × 2 first order quasi-linear hyperbolic PDEs was studied and the invariant graph was introduced.
Abstract: This article is concerned with the design of boundary control laws for stabilizing systems of 2 × 2 first order quasi-linear hyperbolic PDEs. A new graph represen- tation of such systems represents the interactions between the characteristic curves and the boundary control laws, the invariant graph, is introduced. The structure of the invariant graph is used to design stabilizing control laws and an analytical stability condition is given.
TL;DR: This paper proves that every 2 k -ordered (resp.
04 Dec 2004
TL;DR: The experimental results clearly show that the new improvement graph based k-exchange cycle neighborhood improves the accuracy significantly, especially for large scale heuristic search.
Abstract: In this paper, we propose a new neighborhood structure based on the improvement graph for solving the Robust Graph Coloring Problem, an interesting extension of classical graph coloring Different from the traditional neighborhood where the color of only one vertex is modified, the new neighborhood involves several vertices In addition, the questions of how to select the modified vertices and how to modify them are modelled by an improvement graph and solved by a Dynamic Programming method The experimental results clearly show that our new improvement graph based k-exchange cycle neighborhood improves the accuracy significantly, especially for large scale heuristic search.
Journal Article•
TL;DR: In this article, the authors summarise some recent results in extremal graph theory and present some open questions and conjectures, and discuss some open conjectures and open questions in extreme graph theory.
27 Oct 2004
TL;DR: This paper tries to solve the problem of graph partitioning in the multi-objective way by using a population version of the SMOSA algorithm in combination with a diversity preservation method proposed in the SPEA2 algorithm.
Abstract: One significant problem of optimization which occurs in many real applications is that of graph partitioning. It consist of obtaining a partition of the vertices of a graph into a given number of roughly equal parts, whilst ensuring that the number of edges connecting vertices of different sub-graphs is minimized. In the single-objective (traditional) graph partitioning model the imbalance is considered a constraint. However, in same applications it is necessary to extend this model to its multi-objective formulation, where the imbalance is also an objective to minimize. This paper try to solve this problem in the multi-objective way by using a population version of the SMOSA algorithm in combination with a diversity preservation method proposed in the SPEA2 algorithm.
01 Jan 2004
TL;DR: where E(v) is the set of edges that have v as an end-point, and the graph G is vertexmagic if a vertex-magic total labelling of G exists.
Abstract: where E(v) is the set of edges that have v as an end-point. The total labelling λ of G is vertex-magic if every vertex has the same weight, and the graph G is vertexmagic if a vertex-magic total labelling of G exists. Magic labellings of graphs were introduced by Sedlácěk [5] in 1963, and vertex-magic total labellings first appeared in 2002 in [4]. For a dynamic survey of various forms of graph labellings see [1]; for details of vertex-magic graphs, see [6].
TL;DR: The set of cycle lengths occurring in any hamiltonian graph G of order n and maximum degree Δ is described and the stability s(P) for the property of being pancyclic satisfies.
Abstract: Let n and Δ be two integers such that 2≤Δ≤n−1. We describe the set of cycle lengths occurring in any hamiltonian graph G of order n and maximum degree Δ. We conclude that for the case ** this set contains all the integers belonging to the union [3,2Δ−n+2]∪[n−Δ+2,Δ+1], and for ** it contains every integer between 3 and Δ+1. We also study the set of cycle lengths in a hamiltonian graph with two fixed vertices of large degree sum. Our main results imply that the stability s(P) for the property of being pancyclic satisfies **.
Journal Article•
TL;DR: It is shown that every hamiltonian claw-free graph with a vertex x of degree d(x) ≥ 7 has a 2-factor consisting of exactly two cycles.
Abstract: We show that every hamiltonian claw-free graph with a vertex x of degree d(x) ≥ 7 has a 2-factor consisting of exactly two cycles
TL;DR: It is shown that every 3-connected claw-free graph G on at most 5δ−8 vertices is hamiltonian connected, where δ denotes the minimum degree in G.
Abstract: A well-known result by O. Ore is that every graph of order n with d(u)+d(v)≥n+1 for any pair of nonadjacent vertices u and v is hamiltonian connected (i.e., for every pair of vertices, there is a hamiltonian path joining them). In this paper, we show that every 3-connected claw-free graph G on at most 5δ−8 vertices is hamiltonian connected, where δ denotes the minimum degree in G. This result generalizes several previous results.