Additional Vertices Research Articles

Filtered Quadrics for High‐Speed Geometry Smoothing and Clustering

AbstractModern 3D capture pipelines produce dense surface meshes at high speed, which challenge geometric operators to process such massive data on‐the‐fly. In particular, aiming at instantaneous feature‐preserving smoothing and clustering disqualifies global variational optimizers and one usually relies on high‐performance parallel kernels based on simple measures performed on the positions and normal vectors associated with the surface vertices. Although these operators are effective on small supports, they fail at properly capturing larger scale surface structures. To cope with this problem, we propose to enrich the surface representation with filtered quadrics, a compact and discriminating range space to guide processing. Compared to normal‐based approaches, this additional vertex attribute significantly improves feature preservation for fast bilateral filtering and mode‐seeking clustering, while exhibiting a linear memory cost in the number of vertices and retaining the simplicity of convolutional filters. In particular, the overall performance of our approach stems from its natural compatibility with modern fine‐grained parallel computing architectures such as graphics processor units (GPU). As a result, filtered quadrics offer a superior ability to handle a broad spectrum of frequencies and preserve large salient structures, delivering meshes on‐the‐fly for interactive and streaming applications, as well as quickly processing large data collections, instrumental in learning‐based geometry analysis.

Connected domination game

In this paper we introduce a domination game based on the notion of connected domination. Let G = (V,E) be a connected graph of order at least 2. We define a connected domination game on G as follows: The game is played by two players, Dominator and Staller. The players alternate taking turns choosing a vertex of G (Dominator starts). A move of a player by choosing a vertex v is legal, if (1) the vertex v dominates at least one additional vertex that was not dominated by the set of previously chosen vertices and (2) the set of all chosen vertices induces a connected subgraph of G. The game ends when none of the players has a legal move (i.e., G is dominated). The aim of Dominator is to finish as soon as possible, Staller has an opposite aim. Let D be the set of played vertices obtained at the end of the connected domination game (D is a connected dominating set of G). The connected game domination number of G, denoted cg(G), is the minimum cardinality of D, when both players played optimally on G. We provide an upper bound on cg(G) in terms of the connected domination number. We also give a tight upper bound on this parameter for the class of 2-trees. Next, we investigate the Cartesian product of a complete graph and a tree, and we give exact values of the connected game domination number for such a product, when the tree is a path or a star. We also consider some variants of the game, in particular, a Staller-start game.

Secondary Polytope and Secondary Power Diagram

An ingenious construction of Gel’fand, Kapranov, and Zelevinsky [5] geometrizes the triangulations of a point configuration, such that all coherent triangulations form a convex polytope, the so-called secondary polytope. The secondary polytope can be treated as a weighted Delaunay triangulation in the space of all possible coherent triangulations. Naturally, it should have a dual diagram. In this work, we explicitly construct the secondary power diagram, which is the power diagram of the space of all possible power diagrams with nonempty boundary cells. Secondary power diagram gives an alternative proof for the classical secondary polytope theorem based on Alexandrov theorem. Furthermore, secondary power diagram theory shows one can transform a nondegenerated coherent triangulation to another nondegenerated coherent triangulation by a sequence of bistellar modifications, such that all the intermediate triangulations are nondegenerated and coherent. As an application of this theory, we propose an algorithm to triangulate a special class of 3d nonconvex polyhedra without using additional vertices. We prove that this algorithm terminates in $$O({{n}^{3}})$$ time.

DEKOMPOSISI GRAF MATAHARI

Graph theory is a branch of mathematics that facilitates problem solving. There are a lot of researches which concern on this issue. Various kinds of terms are introduced, one of them is graph decomposition. Graph decomposition is sub graphs collection of non-empty G graph {Hi} until Hi = 〈Ei〉 for non-empty sub graph Ei of E (G), where {Ei} is a partition of E (G). Sub graph Hi in decomposition G do not contain of isolated points. If {Hi} is a decomposition of G, it is denoted by .The discussion of graph decomposition can be developed in graph decomposition through various types. One of the types is decomposition of sun graphs. Sun graph is a graph formed from a circle Cn in which each vertex on a circle graph is given one additional vertex with a degree. So, each vertex in sun graph has 3 degrees, except the edge of cortex which only have 1 degree. The sun graph is the result of corona between two graphs, namely a circular graph with n vertex and complement of a complete graph with 1 number of vertex . The sun graph is denoted by where n is the number of vertex in circle graph. If the vertex naming refers to one vertex (with clockwise rules) and additional vertex naming connected to a circle vertex graph (vi), where the additional vertex has a degree of one, then the rule of naming is and sun graph is partitioned into a sub graph H_i = 〈Ei〉 in the form of K 2 where i ≠ j so that H_i∩H_j = ∅, for i = 1,2,3, ..., n with sub graph If every i + 1, i + 2> n has an implicit + 1 and i + 2 will be expressed as an integers 1,2,3, ..., n (mod n), then the sun graph is 2K 2- decomposition. So, for sun graph n ≥3 is 2K2-decomposition. Keywords: Decomposition, Sun Graph .

Sequence Alignment on Directed Graphs.

Genomic variations in a reference collection are naturally represented as genome variation graphs. Such graphs encode common subsequences as vertices and the variations are captured using additional vertices and directed edges. The resulting graphs are directed graphs possibly with cycles. Existing algorithms for aligning sequences on such graphs make use of partial order alignment (POA) techniques that work on directed acyclic graphs (DAGs). To achieve this, acyclic extensions of the input graphs are first constructed through expensive loop unrolling steps (DAGification). Furthermore, such graph extensions could have considerable blowup in their size and in the worst case the blow-up factor is proportional to the input sequence length. We provide a novel alignment algorithm V-ALIGN that aligns the input sequence directly on the input graph while avoiding such expensive DAGification steps. V-ALIGN is based on a novel dynamic programming (DP) formulation that allows gapped alignment directly on the input graph. It supports affine and linear gaps. We also propose refinements to V-ALIGN for better performance in practice. With the proposed refinements, the time to fill the DP table has linear dependence on the sizes of the sequence, the graph, and its feedback vertex set. We conducted experiments to compare the proposed algorithm against the existing POA-based techniques. We also performed alignment experiments on the genome variation graphs constructed from the 1000 Genomes data. For aligning short sequences, standard approaches restrict the expensive gapped alignment to small filtered subgraphs having high similarity to the input sequence. In such cases, the performance of V-ALIGN for gapped alignment on the filtered subgraph depends on the subgraph sizes.

Representation of heat exchanger networks using graph formalism

This contribution addressed the systematic representation of heat exchanger networks thanks to graph formalism. The energy representation of one heat exchanger is presented based on the incidence matrices of the graph related to heat transfer and the graph of the heat convection. Then the global heat exchanger network is obtained from the heat exchanger graph models as well as the incidence matrices of the interconnection obtained by adding additional vertices corresponding to the connector elements. The Port Hamiltonian representation of the heat network exchanger is given in the case of constant pressure. Then we show the incremental passivity of one compartment of the heat exchanger.

Generalised Bagemihl polyhedra and a tight bound on the number of interior Steiner points

We call a 3d polyhedron irreducible if it cannot be decomposed into a set of convex polytopes without adding additional vertices, called Steiner points. Such polyhedra are primitive shapes which are the essence of the reason that will force any 3d convex decomposition algorithm to insert Steiner points in its interior. In this paper, we construct a class of 3d irreducible polyhedra, called generalised Bagemihl polyhedra, with n≥6 vertices. We show that such polyhedra have the same combinatorial structure as the Schönhardt and Bagemihl polyhedra. The most interesting property of these polyhedra is that one can construct it in such a way that it will need more than one interior Steiner point to be triangulated. Given a generalised Bagemihl polyhedron with n≥6 vertices, we show that it can be triangulated by adding at most n−52 interior Steiner points. Moreover, this number is tight. We show an application of using these polyhedra to evaluate the quality of existing 3d boundary recovery tetrahedralisation algorithms.

The relationship between [formula omitted]-forcing and [formula omitted]-power domination

Zero forcing and power domination are iterative processes on graphs where an initial set of vertices are observed, and additional vertices become observed based on some rules. In both cases, the goal is to eventually observe the entire graph using the fewest number of initial vertices. The concept of k-power domination was introduced by Chang et al. (2012) as a generalization of power domination and standard graph domination. Independently, k-forcing was defined by Amos et al. (2015) to generalize zero forcing. In this paper, we combine the study of k-forcing and k-power domination, providing a new approach to analyze both processes. We give a relationship between the k-forcing and the k-power domination numbers of a graph that bounds one in terms of the other. We also obtain results using the contraction of subgraphs that allow the parallel computation of k-forcing and k-power dominating sets.

Continuous average Straightness in spatial graphs

The Straightness is a measure designed to characterize a pair of vertices in a spatial graph. It is defined as the ratio of the Euclidean distance to the graph distance between these vertices. It is often used as an average, for instance to describe the accessibility of a single vertex relatively to all the other vertices in the graph, or even to summarize the graph as a whole. In some cases, one needs to process the Straightness between not only vertices, but also any other points constituting the graph of interest. Suppose for instance that our graph represents a road network and we do not want to limit ourselves to crossroad-to-crossroad itineraries, but allow any street number to be a starting point or destination. In this situation, the standard approach consists in: 1) discretizing the graph edges, 2) processing the vertex-to-vertex Straightness considering the additional vertices resulting from this discretization, and 3) performing the appropriate average on the obtained values. However, this discrete approximation can be computationally expensive on large graphs, and its precision has not been clearly assessed. In this article, we adopt a continuous approach to average the Straightness over the edges of spatial graphs. This allows us to derive 5 distinct measures able to characterize precisely the accessibility of the whole graph, as well as individual vertices and edges. Our method is generic and could be applied to other measures designed for spatial graphs. We perform an experimental evaluation of our continuous average Straightness measures, and show how they behave differently from the traditional vertex-to-vertex ones. Moreover, we also study their discrete approximations, and show that our approach is globally less demanding in terms of both processing time and memory usage. Our R source code is publicly available under an open source license.

On The Partition Dimension of a Lollipop Graph and a Generalized Jahangir Graph

Let G be a connected graph with vertex set V (G), such that V (G) can be divided into any partition set S. The set Π with S ∈ Π is a resolving partition of G if each vertex in G has a distinct representation with respect to Π, and Π is an ordered k−partition. The minimum cardinality of resolving k−partitions of V (G) is called a partition dimension of G, denoted by pd(G). The lollipop graph Lm,n is a graph obtained by joining a complete graph Km to a path Pn with a bridge. A generalized Jahangir graph is a graph consisting of a cycle Cmn and one additional vertex which is adjacent to n vertices of Cmn at m distance to each other on Cmn. Many researchers have conducted research in determining the partition dimension for speci c graph classes. They are as references to determine some of the graph classes that haven’t been studied previously.In this paper, we determine the partition dimension of a lollipop graph Lm,n and a generalized Jahangir graph Jm,n. The research methods in this paper is a book study.The results of this paper are as follows. We obtain the partition dimension of a lollipop graph pd(Lm,n) = m for m ≥ 3 and n ≥ 1. The partition dimension of a generalized Jahangir graph consists of two cases. We showed that pd(Jm,n) = 3 for n = 3, 4, 5 and we prove that .

Some Results on Rosa-type Labelings of Graphs

Labelings that are used in graph decompositions are called Rosa-type labelings. The gamma-labeling of an almost-bipartite graph is a natural generalization of an alpha-labeling of a bipartite graph. It is known that if a bipartite graph G with m edges possesses an alpha-labeling or an almost-bipartite graph G with m edges possesses a gamma-labeling, then the complete graph K_{2mx+1} admits a cyclic G-decomposition. A variation of an alpha-labeling is introduced in this paper by allowing additional vertex labels and some conditions on edge labels and show that whenever a bipartite graph G admits such a labeling, then there exists a supergraph H of G such that H is almost-bipartite and H has a gamma-labeling.

Coloring graphs without fan vertex-minors and graphs without cycle pivot-minors

A fan Fk is a graph that consists of an induced path on k vertices and an additional vertex that is adjacent to all vertices of the path. We prove that for all positive integers q and k, every graph with sufficiently large chromatic number contains either a clique of size q or a vertex-minor isomorphic to Fk. We also prove that for all positive integers q and k≥3, every graph with sufficiently large chromatic number contains either a clique of size q or a pivot-minor isomorphic to a cycle of length k.

Extremal and Degree Conditions for Path Extendability in Digraphs

In the study of cycles and paths, the meta-conjecture of Bondy that sufficient conditions for Hamiltonicity often imply pancyclicity has motivated research on the existence of cycles and paths of many lengths. Hendry further introduced the stronger concepts of cycle extendability and path extendability, which require that every cycle or path can be extended to another one with one additional vertex. These concepts have been studied extensively, but there exist few results on path extendability in digraphs, as far as we know. In this paper, we make the first attempt in this direction. We establish a number of extremal and degree conditions for path extendability in general digraphs. Moreover, we prove that every path of length at least two in a regular tournament is extendable, with some exceptions. One of our proof approaches is a new contraction operation to transform nonextendable paths into nonextendable cycles.

On Tetrahedralisations Containing Knotted and Linked Line Segments

This paper considers a set of twisted line segments in 3d such that they form a knot (a closed curve) or a link of two closed curves. Such line segments appear on the boundary of a family of 3d indecomposable polyhedra (like the Schönhardt polyhedron) whose interior cannot be tetrahedralised without additional vertices added. On the other hand, a 3d (non-convex) polyhedron whose boundary contains such line segments may still be decomposable as long as the twist is not too large. It is therefore interesting to consider the question: when there exists a tetrahedralisation contains a given set of knotted or linked line segments?In this paper, we studied a simplified question with the assumption that all vertices of the line segments are in convex position. It is straightforward to show that no tetrahedralisation of 6 vertices (the three-line-segments case) can contain a trefoil knot. Things become interesting when the number of line segments increases. Since it is necessary to create new interior edges to form a tetrahedralisation. We provided a detailed analysis for the case of a set of 4 line segments. This leads to a crucial condition on the orientation of pairs of new interior edges which determines whether this set is decomposable or not. We then prove a new theorem about the decomposability for a set of n (n ≥ 3) knotted or linked line segments. This theorem implies that the family of polyhedra generalised from the Schonhardt polyhedron by Rambau [1] are all indecomposable.

Stationary states in quantum walk search

When classically searching a database, having additional correct answers makes the search easier. For a discrete-time quantum walk searching a graph for a marked vertex, however, additional marked vertices can make the search harder by causing the system to approximately begin in a stationary state, so the system fails to evolve. In this paper, we completely characterize the stationary states, or 1-eigenvectors, of the quantum walk search operator for general graphs and configurations of marked vertices by decomposing their amplitudes into uniform and flip states. This infinitely expands the number of known stationary states and gives an optimization procedure to find the stationary state closest to the initial uniform state of the walk. We further prove theorems on the existence of stationary states, with them conditionally existing if the marked vertices form a bipartite connected component and always existing if non-bipartite. These results utilize the standard oracle in Grover's algorithm, but we show that a different type of oracle prevents stationary states from interfering with the search algorithm.

Minimum Degrees of Minimal Ramsey Graphs for Almost‐Cliques

AbstractFor graphs F and H, we say F is Ramsey for H if every 2‐coloring of the edges of F contains a monochromatic copy of H. The graph F is Ramsey H‐minimal if F is Ramsey for H and there is no proper subgraph of F so that is Ramsey for H. Burr et al. defined to be the minimum degree of F over all Ramsey H‐minimal graphs F. Define to be a graph on vertices consisting of a complete graph on t vertices and one additional vertex of degree d. We show that for all values ; it was previously known that , so it is surprising that is much smaller. We also make some further progress on some sparser graphs. Fox and Lin observed that for all graphs H, where is the minimum degree of H; Szabó et al. investigated which graphs have this property and conjectured that all bipartite graphs H without isolated vertices satisfy . Fox et al. further conjectured that all connected triangle‐free graphs with at least two vertices satisfy this property. We show that d‐regular 3‐connected triangle‐free graphs H, with one extra technical constraint, satisfy ; the extra constraint is that H has a vertex v so that if one removes v and its neighborhood from H, the remainder is connected.

Drell-Yan process as an avenue to test a noncommutative standard model at the Large Hadron Collider

We study the Drell-Yan process at the Large Hadron Collider in presence of the noncommutative extension of standard model. Using the Seiberg-Witten map, we calculate the production cross-section to the first order in the noncommutative parameter $\Theta_{\mu\nu}$. Although this idea is evolving for long, limited amount of phenomenological analysis was done so far and dominantly in the context of linear collider. Some outstanding feature from this non-minimal noncommutative standard model not only modify the couplings over SM production channel, also allow additional nonstandard vertices which can play a significant role. Hence in the Drell-Yan process, as studied in present analysis, one also needs to account for gluon fusion process at the tree level. Some of the characteristic signatures such as oscillatory azimuthal distributions are outcome of the momentum dependent effective couplings. We explore the noncommutative scale $\Lambda_{NC} \ge 0.4~\rm{TeV}$ considering different machine energy ranging from $7~\rm{TeV}$ to $13~\rm{TeV}$.

M2-brane surface operators and gauge theory dualities in Toda

We give a microscopic two dimensional ${\cal N}=(2,2)$ gauge theory description of arbitrary M2-branes ending on $N_f$ M5-branes wrapping a punctured Riemann surface. These realize surface operators in four dimensional ${\cal N}=2$ field theories. We show that the expectation value of these surface operators on the sphere is captured by a Toda CFT correlation function in the presence of an additional degenerate vertex operator labelled by a representation ${\cal R}$ of $SU(N_f)$, which also labels M2-branes ending on M5-branes. We prove that symmetries of Toda CFT correlators provide a geometric realization of dualities between two dimensional gauge theories, including ${\cal N}=(2,2)$ analogues of Seiberg and Kutasov--Schwimmer dualities. As a bonus, we find new explicit conformal blocks, braiding matrices, and fusion rules in Toda CFT.

Enumeration of chord diagrams on many intervals and their non-orientable analogs

Two types of connected chord diagrams with chord endpoints lying in a collection of ordered and oriented real segments are considered here: the real segments may contain additional bivalent vertices in one model but not in the other. In the former case, we record in a generating function the number of fatgraph boundary cycles containing a fixed number of bivalent vertices while in the latter, we instead record the number of boundary cycles of each fixed length. Second order, non-linear, algebraic partial differential equations are derived which are satisfied by these generating functions in each case giving efficient enumerative schemes. Moreover, these generating functions provide multi-parameter families of solutions to the KP hierarchy. For each model, there is furthermore a non-orientable analog, and each such model likewise has its own associated differential equation. The enumerative problems we solve are interpreted in terms of certain polygon gluings. As specific applications, we discuss models of several interacting RNA molecules. We also study a matrix integral which computes numbers of chord diagrams in both orientable and non-orientable cases in the model with bivalent vertices, and the large-N limit is computed using techniques of free probability.

The Ring Tree Facility Location Problem

In this work we discuss a facility location variant of the capacitated ring tree problem. The new model generalizes vehicle routing problems and Steiner tree problems, yielding applicability in telecommunication and transportation networks. In this ring tree facility location problem (RTFLP) two layers of networks have to be designed to connect two different types of customers to a central depot. The first, or inner layer, consists of cycles that intersect in the depot and collect all type 2 customers, and some of the type 1 customers. An outer layer is represented by a forest that contains the remaining type 1 customers such that each tree shares exactly one vertex with the inner layer. Capacity bounds apply to the number of connected substructures emanating from the depot, the number of customers in each of these so-called ring trees, and in each tree of the forest. Additional optional Steiner vertices can be used to reduce the overall costs, which are layer-dependent edge costs and facility location costs at the vertices in which the two layers coincide. Our contribution is the introduction of the RTFLP, the development of two mathematical formulations, and preliminary computational results for the first RTFLP test set derived from instances from the literature.