Complement Of Graph Research Articles

Algorithms for the clique problem with multiple-choice constraints under a series–parallel dependency graph

The clique problem with multiple-choice constraints (CPMC), i.e. the problem of finding a k -clique in a k -partite graph with known partition, occurs as a substructure in many real-world applications, in particular scheduling and railway timetabling. Although CPMC is NP-complete in general, it is known to be solvable in polynomial time when the so-called dependency graph of G is a forest. In this article, we focus on the special case CPMCSP, where the dependency graph of G is series–parallel. We give a polynomial-time algorithm for CPMCSP using dynamic programming. Further, we provide some facet-inducing inequalities of the CPMCSP polytope, mainly using properties of the stable set polytope of the complement graph of G . Among these, we give a separation algorithm for the so-called embedded odd-clique-cycle inequalities using dynamic programming. If the number of vertices per subset of the k -partition is bounded, then its runtime is polynomial in the size of the dependency graph.

On Connectivity of Zero-Divisor Graphs and Complement Graphs of some Semi-Local Rings

Zero-divisor graphs have been a key area of focus for many researchers. For the semi local ring R of finite cartesian product of finite fields, we consider the zero divisor graph of R denoted by Γ(R) with vertex set as the non-zero zero-divisors of R where two vertices u and v are adjacent if and only if the product of u and v is the additive identity of the Ring R. The objective of this paper is to determine the number of cut vertices and cut edges, vertex connectivity and edge connectivity of the zero divisor graph Γ(R) and complement graph Γ(R).

Erratum: The distance spectrum of the complements of graphs of diameter greater than three

Erratum: The distance spectrum of the complements of graphs of diameter greater than three

Data-driven optimal sensor placement for high-dimensional system using annealing machine

We propose a novel method for solving optimal sensor placement problem for high-dimensional system using an annealing machine. The sensor points are calculated as a maximum clique problem of the graph, the edge weight of which is determined by the proper orthogonal decomposition mode obtained from data based on the fact that a high-dimensional system usually has a low-dimensional representation. Since the maximum clique problem is equivalent to the independent set problem of the complement graph, the independent set problem is solved using Fujitsu Digital Annealer. In contrast to existing greedy methods, which select the optimal point at each step and never reconsider the point selected previously, the proposed method is superior because it is able to find the optimal set of points. As a demonstration of high dimensional system, the pressure distribution measured by the pressure-sensitive paint method, which is an optical flow diagnose method, is reconstructed from the pressure data at the calculated sensor points. The root mean square errors (RMSEs) between the pressures measured by pressure transducers and the pressures reconstructed from the proposed method, an existing greedy method, and random selection method are compared. The similar RMSE is achieved by the proposed method using approximately 1/5 number of sensor points calculated by the existing method. This method is of great importance as a novel approach for optimal sensor placement problem and a new engineering application of an annealing machine.

Complexity results on open-independent, open-locating-dominating sets in complementary prism graphs

For a finite, simple, undirected graph G = ( V ( G ) , E ( G ) ) , an open-dominating set S ⊆ V ( G ) is such that every vertex in G has at least one neighbor in S . An open-independent, open-locating-dominating set S ⊆ V ( G ) ( O L D o i n d -set for short) is such that no two vertices in G have the same set of neighbors in S and each vertex in S is open-dominated exactly once by S . The problem of deciding whether or not G has such set is known to be NP -complete. The complementary prism of G is the graph G G ¯ , formed from the disjoint union of G and its complement G ¯ by adding the edges of a perfect matching between the corresponding vertices of G and G ¯ . Various properties, upper bounds and logarithmic lower bounds on the sizes of minimal O L D o i n d -sets in complementary prisms are presented. We show that deciding, for a given graph G , whether or not G G ¯ has an O L D o i n d -set is an NP -complete problem. On the other hand, if the girth of G is at least four, we show that the O L D o i n d -set of G G ¯ , if it exists, can be found in polynomial time.

Partially balanced incomplete block (PBIB)-designs arising from diametral paths in some strongly regular graphs

The partially balanced incomplete block (PBIB)-designs and association schemes arising from different graph parameters is a well-studied concept. In this paper, we define and construct PBIB-designs with association schemes in strongly regular graphs through the nodes belonging to the diametral paths as blocks. A [Formula: see text]-design, called a diametral-design (in short) over a strongly regular graph [Formula: see text] of degree [Formula: see text], diameter [Formula: see text], is an ordered pair [Formula: see text], where [Formula: see text] and [Formula: see text] is the set of all diametral paths of [Formula: see text], called blocks, containing the nodes belonging to the diametral paths, satisfying the following conditions: (i) If [Formula: see text] and [Formula: see text], then there are exactly [Formula: see text] blocks containing [Formula: see text]; (ii) If [Formula: see text], [Formula: see text] and [Formula: see text], then there are exactly [Formula: see text]-blocks containing [Formula: see text]. We construct PBIB-designs with two-association schemes through diametral paths corresponding to the strongly regular graphs such as the square lattice graphs [Formula: see text], the triangular graphs [Formula: see text], the three Chang graphs, the Petersen graph, the Shrikhande graph, the Clebsch graph, the complement of Clebsch graph, the complement of Schläfli graph, complete bipartite graphs, the Hoffman–Singleton graph, etc. and in each case we give the composition of the diametral paths. These serve as extensions for the collection of near impossible class of strongly regular graphs with given parameters having two-association schemes.

Computing Forgotten Eccentric Topological Index of Complement of Line Graphs

The forgotten topological index of a graph G is defined as the sum of the cube of the degrees of its vertices. In this paper we establish the forgotten eccentric topological index of complement of line graphs and it is denoted by . These are defined as (v).e(v), where (edge vertex degree) and establish some upper and lower bounds for them.

On the clique behavior and Hellyness of the complements of regular graphs

A collection of sets is intersecting, if any pair of sets in the collection has nonempty intersection. A collection of sets has the Helly property if any intersecting subcollection has nonempty intersection. A graph G is Helly if the collection of maximal complete subgraphs of G has the Helly property. We prove that if G is a k-regular graph with n vertices such that n>3k+2k2−k, then the complement G‾ is not Helly. We also consider the problem of whether the properties of Hellyness and convergence under the clique graph operator are equivalent for the complement of k-regular graphs, for small values of k.

Total Coloring of Some Classes of Cayley Graphs on Non-Abelian Groups

Total Coloring of a graph G is a type of graph coloring in which any two adjacent vertices, an edge, and its incident vertices or any two adjacent edges do not receive the same color. The minimum number of colors required for the total coloring of a graph is called the total chromatic number of the graph, denoted by χ″(G). Mehdi Behzad and Vadim Vizing simultaneously worked on the total colorings and proposed the Total Coloring Conjecture (TCC). The conjecture states that the maximum number of colors required in a total coloring is Δ(G)+2, where Δ(G) is the maximum degree of the graph G. Graphs derived from the symmetric groups are robust graph structures used in interconnection networks and distributed computing. The TCC is still open for the circulant graphs. In this paper, we derive the upper bounds for χ″(G) of some classes of Cayley graphs on non-abelian groups, typically Cayley graphs on the symmetric groups and dihedral groups. We also obtain the upper bounds of the total chromatic number of complements of Kneser graphs.

Majority Dom-Chromatic Number of a Bipartite Graph

In this article, the majority dom-chromatic sets of a bipartite graphs are studied. The characterization theorems on the majority dom-chromatic number γMχ (G) for bipartite graphs are determined. Also its relationship with other graph theoretic parameters and the majority dom-chromatic number for complement of a bipartite graphs are investigated.

Journal of Pharmaceutical Negative Results ¦ Volume 13 ¦ Special Issue 6 ¦ 2022 1817 The Strongly Rainbow Connected Number of the Inverse Graphs

The two key parameters in this study of graph theory are the rainbow connected number and strongly rainbow connected numbers. The purpose of this study is to identify the strongly rainbow connected numbers of the inverse graphs corresponding to the modular groups under various operations, and to evaluate and explain the characteristics of these numbers in various contexts.The objective of this Study is to apply these concepts and results in the field of Networking and Coding. An edge colored graph G is called rainbow-connected if a path whose edges have different colors that connects any two vertices. The minimum k for which there exist a rainbow k−coloring of G is called the rainbow connection number of G, denoted by rc(G), the adjacent edges may be allowed to color with the same color. The graph G is strongly rainbow connected if there exists a rainbow u-v geodesic for every pair of vertices u and v in G.

Chromatic Index and Line graphs of Neighbourly Irregular Fuzzy Chemical Graphs(GNIF C ) among s-block and p-block elements

In this paper, we have worked on fuzzy vertex coloring and fuzzy edge coloring for neighbourly irregular chemical graph GNIC and its complement graph ~G NIC: Also derived line graphs L(GNIC) of neighbourly irregular chemical graph and line graph of complement neighbourly irregular chemical graph (L(~G NIC)): Using fuzzy edge coloring, we have find chromatic index for »(GNIFC); irregular chromatic index »ir(GNIFC) also for »irL(GNIFC) and »irL(~G NIFC): We have given the required propositions with the required examples for each. And we fund strength of the »L(GNIFC) = (N;W); and the colors.

Temporal attention aware dual-graph convolution network for air traffic flow prediction

Air traffic flow prediction is vital for its supporting function for collaborative decision making in Air Traffic Management. However, due to the inherent spatial and temporal dependencies of air traffic flow and the irregular sector structure in which flow operates in, it is still a challenging problem. To solve this problem, numerous methods are proposed considering airspace adjacency, while flight routes and the origin-destination dependency are not taken into account. In this paper, we propose a temporal attention aware dual-graph convolution network (TAaDGCN) to predict air traffic flow, in which the airspace structure and routes of flow are both included. Firstly, a complementary spatial dual graph convolution module is constructed to capture the dependencies of adjacent sectors and origin-destination (OD) sectors. Then, to include long path information, a spatial embedding (SE) block is adopted to represent potential related sectors of flight traversal. Furthermore, to characterize temporal evolution pattern, a temporal attention (TA) module is applied to access past features of input sequence. Based on the blocks stated above, a spatio-temporal block is constructed in which multiple spatial and temporal dependencies are covered. The experimental results on real-world flight data demonstrate the proposed method can achieve a better prediction performance than other state-of-the-art comparison methods, especially superior to the methods that ignore the sector spatial structure.

The distance spectrum of the complements of graphs with two pendent vertices

The distance spectrum of the complements of graphs with two pendent vertices

Domination number and feedback vertex number of complements of line graphs

In general, determining the domination number and the feedback vertex number of a claw-free graph (even a line graph) is NP-hard. In contrast, the situation becomes different for the complement of a line graph. In this paper, it is shown that for a claw-free G with and thus determining the domination number of the complement of a claw-free G with is polynomial, where is the independence number of G. Furthermore, if a graph G is not a star, has no isolated vertex and isolated edge, then where J(G) is the complement of the line graph L(G) of G. If a graph G is not a star, and has no isolated vertex, provided For the case when is also given. Thereby, we are able to show that determining is polynomial.

Efficient complementary graph convolutional network without negative sampling for item recommendation

Learning vector representations (a.k.a., embeddings) of users and items lies at the heart of building a modern recommender system. Typically, graph neural networks (GNN) have been a prevalent way to learn user and item embeddings with implicit feedback. Nevertheless, since item popularity in many recommendation scenarios follows a long-tail distribution, we argue that recent GNN-based recommenders are not sufficient to yield satisfactory embeddings, especially for tail items. More importantly, most of GNN-based methods depend on negative sampling in model optimization, which makes them inefficient in the GPU era and highly sensitive to the design of negative sampler. In light of these limitations, we propose a novel recommendation method named ECGN, short for efficient complementary graph convolutional network. Specifically, by means of the complementary relationships amongst items, we introduce a complementary-aware graph convolution module, which better alleviates the long-tail problem. By regarding user behaviors and item relationships as two bipartite graphs, such module could explicitly explore the propagation process of user interest and complementary influence. Furthermore, an item-specific attention mechanism is designed to fuse item representations from different bipartite graphs. To make our solution attractive in large-scale industrial scenarios, we devise an efficient non-sampling learning paradigm to optimize ECGN’s parameters. Experiments on six real-world datasets demonstrate that our method consistently and significantly outperforms several state-of-the-art methods in model effectiveness and training efficiency. Our implementations and datasets are available at https://github.com/wubinzzu/ECGN.

The distance spectrum of the complements of graphs of diameter greater than three

Suppose G is a connected simple graph with the vertex set \(V( G ) = \{ v_1,v_2,\cdots ,v_n \} \). Let \(d_G( v_i,v_j ) \) be the least distance between \(v_i\) and \(v_j\) in G. Then the distance matrix of G is \(D( G ) =( d_{ij} ) _{n\times n}\), where \(d_{ij}=d_G( v_i,v_j ) \). Since D(G) is a non-negative real symmetric matrix, its eigenvalues can be arranged \(\lambda _1(G)\ge \lambda _2(G)\ge \cdots \ge \lambda _n(G)\), where eigenvalues \(\lambda _1(G)\) and \(\lambda _n(G)\) are called the distance spectral radius and the least distance eigenvalue of G, respectively. In this paper, we characterize the unique graph whose distance spectral radius attains maximum and minimum among all complements of graphs of diameter greater than three, respectively. Furthermore, we determine the unique graph whose least distance eigenvalue attains minimum among all complements of graphs of diameter greater than three.

The complement of the intersection graph of ideals of a poset

Let [Formula: see text] be an atomic poset with the least element [Formula: see text]. The complement of the intersection graph of ideals of [Formula: see text], denoted by [Formula: see text], is defined to be a graph whose vertices are all non-trivial ideals of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we consider the complement of the intersection graph of ideals of a poset. We prove that [Formula: see text] is totally disconnected or [Formula: see text], where [Formula: see text] is the set of all isolated vertices of [Formula: see text]. We show that [Formula: see text]. Also, we characterize all posets whose complement of the intersection graph is forest, unicyclic or complete [Formula: see text]-partite graph. Among other results, we prove that [Formula: see text] is weakly perfect; and it is perfect if and only if [Formula: see text]. Finally, we show that [Formula: see text] is class [Formula: see text], where [Formula: see text].

Nordhaus-gaddum inequalities for anti fuzzy graph

The objective of this paper is to finds the lower and upper bounds of Nordhaus-Gaddum inequalities of fuzzy chromatic number for anti-fuzzy graph. This paper analyzes the chromatic index of complementary anti fuzzy graphs in some cases. A theorem is proved for anti-fuzzy graph to be k-critical. Examples are provided to derive the vertex coloring of these graphs.

Solving the inverse graph model for conflict resolution using a hybrid metaheuristic algorithm

The paper is concerned with one of the most important questions in conflict analysis: How decision makers can strategically interact in a conflict to reach a specified equilibrium? More specifically, an inverse preference optimization model is formulated to solve the problem of adjusting the preferences of decision makers so as to make a specified state an equilibrium. To this end we define an algorithm which incorporates the hybridization of particle swarm optimization and genetic algorithm. The proposed algorithm adopts a random neighbor strategy for population initialization, the elite reservation and mixed selection operation, and a diversity strategy for population update to improve its efficiency and effectiveness when searching for the required preferences. This approach can help decision makers or third parties to focus their resources on guiding them toward preferences that lead to a specified resolution. Finally, a real-world dispute over exporting bulk water from Eastern Canada is used to demonstrate the applicability and effectiveness of the approach.