Free Vertices Research Articles

A Graph-Based Hybrid Reconfiguration Deformation Planning for Modular Robots.

The self-reconfigurable modular robotic system is a class of robots that can alter its configuration by rearranging the connectivity of their component modular units. The reconfiguration deformation planning problem is to find a sequence of reconfiguration actions to transform one reconfiguration into another. In this paper, a hybrid reconfiguration deformation planning algorithm for modular robots is presented to enable reconfiguration between initial and goal configurations. A hybrid algorithm is developed to decompose the configuration into subconfigurations with maximum commonality and implement distributed dynamic mapping of free vertices. The module mapping relationship between the initial and target configurations is then utilized to generate reconfiguration actions. Simulation and experiment results verify the effectiveness of the proposed algorithm.

Symposium Title: Hypermirroring- Emotionally Gifted- Living From the Heart- A New Pattern in Children- Supported by Clinical and EEG and BrainMap Data

This paper studies a variation of domination in graphs called (F,B,R)-domination. Let G=(V,E) be a graph and V be the disjoint union of F, B, and R, where F consists of free vertices, B consists of bound vertices, and R consists of required vertices. An (F,B,R)-dominating set of G is a subset D⊆V such that R⊆D and each vertex in B−D is adjacent to some vertex in D. An (F,B,R)-2-stable set of G is a subset S⊆B such that S∩N(R)=0̸ and every two distinct vertices x and y in S have distance d(x,y)>2. We prove that if G is strongly chordal, then αF,B,R,2(G)=γF,B,R(G)−|R|, where γF,B,R(G) is the minimum cardinality of an (F,B,R)-dominating set of G and αF,B,R,2(G) is the maximum cardinality of an (F,B,R)-2-stable set of G. Let D1→∗D2 denote D1 being transferable to D2. We prove that if G is a connected strongly chordal graph in which D1 and D2 are two (F,B,R)-dominating sets with |D1|=|D2|, then D1→∗D2. We also prove that if G is a cactus graph in which D1 and D2 are two (F,B,R)-dominating sets with |D1|=|D2|, then D1∪{1⋅extra}→∗D2∪{1⋅extra}, where ∪{1⋅extra} means adding one extra element.

Artificial Intelligence Knowledge Graph for Dynamic Networks: An Incremental Partition Algorithm

The quick and intelligent requests and answers in artificial intelligence (AI) are inseparable from intelligent data. Knowledge graph makes data more intelligent by establishing association among data, which provides convenience for intelligent search, reasoning and analysis of data. Resource Description Framework (RDF) is an effective data representation model of knowledge graph. This paper takes RDF as the research object and proposes an incremental partition method of intelligent data (IPID) to realize the distributed storage of large-scale AI data. First, IPID gives a mixed object function integrating edge cut and load balancing. Second, IPID devises the initial and incremental partitioning algorithms of RDF. The initial partition divides the original RDF graph into kernel vertices, boundary vertices and free vertices. The boundary and freedom nodes select the kernel vertex with the maximum gain of object function to form a sub-partition. And the incremental partition is in charge of the selection of sub-partition of new and deleted data by the object function. Meanwhile, the incremental partition algorithm would also execute a dynamic adjustment strategy at a certain time interval according to the balance and tightness of sub-partition to satisfy the partitioning object. Finally, IPID is tested on the knowledge graph datasets. The experimental results show that the object function guarantees the quality of knowledge graph partition in edge cut and load balancing, and effectively realizes the incremental partition.

A Deviation-Based Dynamic Vertex Reordering Technique for 2D Mesh Quality Improvement

We propose a novel deviation-based vertex reordering method for 2D mesh quality improvement. We reorder free vertices based on how likely this is to improve the quality of adjacent elements, based on the gradient of the element quality with respect to the vertex location. Specifically, we prioritize the free vertex with large differences between the best and the worst-quality element around the free vertex. Our method performs better than existing vertex reordering methods since it is based on the theory of non-smooth optimization. The downhill simplex method is employed to solve the mesh optimization problem for improving the worst element quality. Numerical results show that the proposed vertex reordering techniques improve both the worst and average element, compared to those with existing vertex reordering techniques.

Mutual transferability for [formula omitted]-domination on strongly chordal graphs and cactus graphs

This paper studies a variation of domination in graphs called (F,B,R)-domination. Let G=(V,E) be a graph and V be the disjoint union of F, B, and R, where F consists of free vertices, B consists of bound vertices, and R consists of required vertices. An (F,B,R)-dominating set of G is a subset D⊆V such that R⊆D and each vertex in B−D is adjacent to some vertex in D. An (F,B,R)-2-stable set of G is a subset S⊆B such that S∩N(R)=0̸ and every two distinct vertices x and y in S have distance d(x,y)>2. We prove that if G is strongly chordal, then αF,B,R,2(G)=γF,B,R(G)−|R|, where γF,B,R(G) is the minimum cardinality of an (F,B,R)-dominating set of G and αF,B,R,2(G) is the maximum cardinality of an (F,B,R)-2-stable set of G. Let D1→∗D2 denote D1 being transferable to D2. We prove that if G is a connected strongly chordal graph in which D1 and D2 are two (F,B,R)-dominating sets with |D1|=|D2|, then D1→∗D2. We also prove that if G is a cactus graph in which D1 and D2 are two (F,B,R)-dominating sets with |D1|=|D2|, then D1∪{1⋅extra}→∗D2∪{1⋅extra}, where ∪{1⋅extra} means adding one extra element.

NEW CONCEPTS OF REGULAR INTERVAL-VALUED FUZZY GRAPHS

Recently, interval-valued fuzzy graph is a growing research topic as it is the generalization of fuzzy graphs. The interval-valued fuzzy graphs are more flexible and compatible than fuzzy graphs due to the fact that they allowed the degree of membership of a vertex to an edge to be represented by interval values in [0.1] rather than the crisp values between 0 and 1. In this paper, we introduce the concepts of regular and totally regular interval-valued fuzzy graphs and discusses some properties of the <TEX>${\mu}$</TEX>-complement of interval-valued fuzzy graph. Self <TEX>${\mu}$</TEX>-complementary interval-valued fuzzy graphs and self-weak <TEX>${\mu}$</TEX>-complementary interval-valued fuzzy graphs are defined and a necessary condition for an interval valued fuzzy graph to be self <TEX>${\mu}$</TEX>-complementary is discussed. We define busy vertices and free vertices in interval valued fuzzy graph and study their image under an isomorphism.

Some isomorphic properties of m-polar fuzzy graphs with applications.

The theory of graphs are very useful tool in solving the combinatorial problems in different areas of computer science and computational intelligence systems. In this paper, we present a frame work to handle m-polar fuzzy information by combining the theory of m-polar fuzzy sets with graphs. We introduce the notion of weak self complement m-polar fuzzy graphs and establish a necessary condition for m-polar fuzzy graph to be weak self complement. Some properties of self complement and weak self complement m-polar fuzzy graphs are discussed. The order, size, busy vertices and free vertices of an m-polar fuzzy graphs are also defined and proved that isomorphic m-polar fuzzy graphs have same order, size and degree. Also, we have presented some results of busy vertices in isomorphic and weak isomorphic m-polar fuzzy graphs. Finally, a relative study of complement and operations on m-polar fuzzy graphs have been made. Applications of m-polar fuzzy graph are also given at the end.

Some properties of vague graphs with application

Theoretical concepts of graphs are highly utilized by computer science applications. Especially in research areas of computer science such as data mining, image segmentation, clustering, image capturing and networking. The vague graphs are more flexible and compatible than fuzzy graphs due to the fact that they have many applications in networks. In this paper, busy vertices, free vertices, strong arc, busy value, isolated vertex, fully free vertex and partial free vertex in vague graphs are defined. Likewise, some interesting properties of isomorphism and weak isomorphism on them are introduced. Finally, we give an application of vague digraph in vulnerability assessment of gas pipeline networks.

Properties of interval valued intuitionistic (S,T) – Fuzzy graphs

The basis of the concept of interval valued intuitionistic fuzzy sets was introduced by K. Atanassov. Interval valued intuitionistic models provide more precision, flexibility, and compatibility to a system than do classic fuzzy models. In this paper, three new types of product operations (direct product, lexicographic product, and strong product) of interval valued intuitionistic (S,T)–fuzzy graphs are defined. One of the most studied classes of fuzzy graphs are regular fuzzy graphs, which appear in many contexts. For example, r-regular fuzzy graphs with connectivity and edge-connectivity equal to r play a key role in designing reliable communication networks. Hence, we introduced the concepts of regular and totally regular interval valued intuitionistic (S,T)–fuzzy graphs. Likewise, we defined busy vertices and free vertices in interval valued intuitionistic (S,T)–fuzzy graphs and studied their images under an isomorphism, which are highly important in fuzzy social networks.

Dynamic Programming on Nominal Graphs

Many optimization problems can be naturally represented as (hyper) graphs, where vertices correspond to variables and edges to tasks, whose cost depends on the values of the adjacent variables. Capitalizing on the structure of the graph, suitable dynamic programming strategies can select certain orders of evaluation of the variables which guarantee to reach both an optimal solution and a minimal size of the tables computed in the optimization process. In this paper we introduce a simple algebraic specification with parallel composition and restriction whose terms up to structural axioms are the graphs mentioned above. In addition, free (unrestricted) vertices are labelled with variables, and the specification includes operations of name permutation with finite support. We show a correspondence between the well-known tree decompositions of graphs and our terms. If an axiom of scope extension is dropped, several (hierarchical) terms actually correspond to the same graph. A suitable graphical structure can be found, corresponding to every hierarchical term. Evaluating such a graphical structure in some target algebra yields a dynamic programming strategy. If the target algebra satisfies the scope extension axiom, then the result does not depend on the particular structure, but only on the original graph. We apply our approach to the parking optimization problem developed in the ASCENS e-mobility case study, in collaboration with Volkswagen. Dynamic programming evaluations are particularly interesting for autonomic systems, where actual behavior often consists of propagating local knowledge to obtain global knowledge and getting it back for local decisions.

Synthèse et étude structurale de Na9Cr(MoO4)6.

The title compound, nona-sodium chromium(III) hexa-kis[molybdate(VI)], Na9Cr(MoO4)6 was prepared by solid-state reactions. The basic structure units are isolated polyhedral clusters composed of a central CrO6 octa-hedron sharing vertices with six MoO4 tetrahedra to form an open framework in which the Na(+) cations are bound to the free vertices of the MoO4 tetra-hedra. The Cr(3+) cation has site symmetry of 32 (6a), one Na atom sits on a twofold axis (18e), and all other atoms are at general positions. The bond-valance-sum model confirms the expected values of ion charges. The title compound is isotypic with Na9Sc(MoO4)6 and Na9Fe(MoO4)6. It is compared and discussed with similar structures.

Bipolar Fuzzy Graphs with Categorical Properties

AbstractTheoretical concepts of graphs are highly utilized by computer science applications. Especially in research areas of computer science such as data mining, image segmentation, clustering, image capturing and networking. In this paper, we discussed some properties of the µ–complement of bipolar fuzzy graphs. Self µ–complement bipolar fuzzy graphs and self weak µ–complement bipolar fuzzy graphs are defined and a necessary condition for a bipolar fuzzy graph to be self µ–complement is given. We defined busy vertices and free vertices in bipolar fuzzy graphs and studied their image under an isomorphism. Categorical properties of bipolar fuzzy graphs are discussed. Also, we investigated some properties of isomorphism on bipolar fuzzy graphs.

A Hölder-type inequality on a regular rooted tree

We establish an inequality which involves a non-negative function defined on the vertices of a finite m-ary regular rooted tree. The inequality may be thought of as relating an interaction energy defined on the free vertices of the tree summed over automorphisms of the tree, to a product of sums of powers of the function over vertices at certain levels of the tree. Conjugate powers arise naturally in the inequality, indeed, Hölder's inequality is a key tool in the proof which uses induction on subgroups of the automorphism group of the tree.

The number of [formula omitted]-free vertices on 3-partite tournaments

Let T be a 3-partite tournament. We say that a vertex v is C 3 ⃗ -free if v does not lie on any directed triangle of T . Let F 3 ( T ) be the set of the C 3 ⃗ -free vertices in a 3-partite tournament and f 3 ( T ) its cardinality. In this paper we prove that if T is a regular 3-partite tournament, then F 3 ( T ) must be contained in one of the partite sets of T . It is also shown that for every regular 3-partite tournament, f 3 ( T ) does not exceed n 9 , where n is the order of T . On the other hand, we give an infinite family of strongly connected tournaments having n − 4 C 3 ⃗ -free vertices. Finally we prove that for every c ≥ 3 there exists an infinite family of strongly connected c -partite tournaments, D c ( T ) , with n − c − 1 C 3 ⃗ -free vertices.

On symmetric configurations in some problems on extremal decomposition. II

Some problems on extremal decomposition in families of nonoverlapping domains containing systems of biangles with free vertices on a circle are considered. Simultaneously, some progress in solving the classical problem on the maximum of a well-known conformal invariant is achieved. This exhibits the role of symmetric configurations in extremal problems under consideration. Bibliography: 11 titles.

Functional [6]Pericyclynes: Synthesis through [14+4] and [8+10] Cyclization Strategies

Critical analysis of possible strategies for the synthesis of novel carbo-benzene derivatives suggests several [(18-n)+n] routes for the preparation of hexaoxy[6]pericyclyne precursors. Beyond the previously attempted [9+9] symmetrical scheme (n=9), the a priori most selective strategies are those for which n=1, 4, 7, 10, 13, and 16. They involve a cyclizing double-propargylation of a C(18-n) omega-bis-terminal-skipped oligoyne containing (19-n)/3 triple bonds with a C(n) omega-dicarbonyl-skipped oligoyne containing (n-1)/3 triple bonds. To complement the previously used [11+7] strategy, the [14+4] and [8+10] strategies were thus explored. They proved to be efficient, affording seven novel hexaoxy[6]pericyclynes corresponding to six different substitution patterns. These compounds were obtained in 7-15 steps as mixtures of stereoisomers. Thus, by using dibenzoylacetylene as the C(4) electrophile, a [14+4] strategy allowed the synthesis of two hexaphenyl representatives with two or four free carbinol vertices. This approach also afforded tetraphenyl representatives in which the two remaining carbinoxy vertices were substituted with either two alkynyl or one 4-anisyl and one 4-pyridyl groups. By using the hexacarbonyldicobalt complex of butynedial as the C(4) electrophile, a [14+4] strategy also allowed the isolation of a tetraphenylhexaoxy[6]pericyclyne with two adjacent unsubstituted carbinol vertices. A regioisomer with two opposite unsubstituted carbinol vertices was obtained through an [8+10] strategy and its oxidation afforded the corresponding pericyclynedione. Several attempts at synthesizing diphenylhexaoxy[6]pericyclynes with four unsubstituted carbinoxy vertices are described. Only an [8+10] strategy allowed the generation of a fragile diphenylhexaoxy[6]pericyclyne with four adjacent unsubstituted carbinoxy vertices, which could be partly characterized. These results show that the synthesis of the nonsubstituted hexahydroxy[6]pericyclyne, the ring carbo-mer of [6]cyclitol, is a difficult challenge.

Lanthanido ammonium cations [NM4]9+ as main structural features in lanthanide(III) nitride chalcogenides and their derivatives

All lanthanide(III) nitride chalcogenides contain lanthanido ammonium cations [NM4]9+ (M = La–Lu) as main structural features. They can occur isolated ({[NM4]9+}∞0, N:M = 1:4) as in compounds of the compositions M4NCh3X3 (Ch = S, Se; X = Cl, Br) and NaM10NS14 or form discrete [N2M6]12+ units (N:M = 1:3) as a pair of tetrahedra with a common edge in the M5NSe6-type representatives. With the same stoichiometry (N:M = 1:3), but a one-dimensional infinite solution {[NM3]6+}∞1, they share two vertices to build up chains in the formula types M3NCh3 (Ch = S, Se) and CsM6N2Te7. By sharing two edges (cis-oriented in M4N2Ch3 (Ch = S, Se), trans-oriented in M4N2Te3 and M5N2Se4Br), chains of the composition {[NM2]3+}∞1 (N:M = 1:2) are generated, but also the condensation via one fused edge and two common corners can lead to chains of the same stoichiometry ({[NM2]3+}∞1 in M4N2S3), when pairs of edge-shared bitetrahedra ([N2M6]12+) join with their four free vertices. The latter way of interconnection is used to construct even porous layers {[NM2]3+}∞2 in the nitride chalcogenides M4N2Ch3 with four-and-eight- (Ch = S) or double-six-ring pores (Ch = Se). Two different types of chains (a cis-edge shared one and a one edge plus two corners shared one, both commensurate by translation) characterize the structure of the nitride chalcogenide halogenides with the formula M6N3S4X (X = Cl, Br), without changing the overall composition (N:M = 1:2). When two chains of cis-edge shared [NM4]9+ tetrahedra are connected by an extra edge per unit, a complex strand with the composition {[N2M3]3+}∞1 (N:M = 1:1.5) emerges as in the mixed anion compounds M3N2SeBr. Almost the same {[N2M3]3+}∞1-chain feature is exhibited by one of the two different chains in compounds such as M5N3S2Cl2, where the other one displays the motif of the pairs of edge-shared bitetrahedra ([N2M6]12+) interconnected via their four free vertices ({[NM2]3+}∞1).

On k-Dependent Domination in Graphs

A subset D of vertices in a graph G is k-dependent if the maximum degree of a vertex in the subgraph 〈D〉 induced by D is at most k. The k-dependent domination number γk(G) of a graph G is the minimum cardinality of a k-dependent dominating set of G. Any k-dependent dominating set D of a graph G with |D|=γk(G) is called a γk-set of G. A vertex x of a graph G is called: (i) γk-good if x belongs to some γk-set, (ii) γk-fixed if x belongs to every γk-set, (iii) γk-free if x belongs to some γk-set but not to all γk-sets, (iv) γk-bad if x belongs to no γk-set. In this paper we deal with γk-good/bad/fixed/free vertices and present results on changing and unchanging of the k-dependent domination number when a graph is modified by adding an edge or deleting a vertex.

Barycentric systems and stretchability

Using a general resolution of barycentric systems we give a generalization of Tutte's theorem on convex drawing of planar graphs. We deduce a characterization of the edge coverings into pairwise non-crossing paths which are stretchable: such a system is stretchable if and only if each subsystem of at least two paths has at least three free vertices (vertices of the outer face of the induced subgraph which are internal to none of the paths of the subsystem). We also deduce that a contact system of pseudo-segments is stretchable if and only if it is extendible.

As-rigid-as-possible shape manipulation

We present an interactive system that lets a user move and deform a two-dimensional shape without manually establishing a skeleton or freeform deformation (FFD) domain beforehand. The shape is represented by a triangle mesh and the user moves several vertices of the mesh as constrained handles. The system then computes the positions of the remaining free vertices by minimizing the distortion of each triangle. While physically based simulation or iterative refinement can also be used for this purpose, they tend to be slow. We present a two-step closed-form algorithm that achieves real-time interaction. The first step finds an appropriate rotation for each triangle and the second step adjusts its scale. The key idea is to use quadratic error metrics so that each minimization problem becomes a system of linear equations. After solving the simultaneous equations at the beginning of interaction, we can quickly find the positions of free vertices during interactive manipulation. Our approach successfully conveys a sense of rigidity of the shape, which is difficult in space-warp approaches. With a multiple-point input device, even beginners can easily move, rotate, and deform shapes at will.