Set Of Graphs Research Articles

Molecular structures of eight hydrogen bond-mediated minoxidil adducts from different aryl acids

Cocrystallization experiments between the widely popular active pharmaceutical ingredient (API) minoxidil and an array of aryl acids coformers were carried out, successfully yielding 8 multi-component samples consisting of 3 co-crystals and 5 salts. Most of them are hydrates. The nature of the samples have been featured by XRD, IR and EA, the melting points were also gauged. The structural and supramolecular aspects are investigated in detail. In all adducts the N-oxide shows various nonbonding contacts. The Hmnd dimer was established at both 1 and 4 by a pair of symmetric NH···O H-bonds. In all cases the minoxidil and coformers were connected by the NH···O/OH···O synthon which is proved to be the highest extensive associates (appearing in 8/8 structures), even in the coexistence of other potentially disruptive non-covalent bonding moieties. Interestingly, the piperidine moieties as effective H-bond acceptor never participated in accepting the H-bonds. The pyrimidine amino also only acts as the H-bond donors.The outcomes unveil that the crystal packing is relied on the strong NH⋯O/OH···O H-bonds. Deep insight into the crystal packing uncovered that a different set of extra ON/OO, CH···C, NH···C, CH···O/CH2O, CHCH/CH2CH/CH2CH2, CH2Cl, CH3-π/CH2-π, NH-π, O-π, and π-π contacts contribute to the stabilization and expansion of the total structures. For the synergism of the various nonbonding forces there had the homo/hetero supramolecular synthons. Some classical graph set notations of R22(9) and R42(8) were present in the relative high frequency.

Bondage and non-bondage sets in regular intuitionistic fuzzy graphs

The concept of strong edges in domination set and its properties are discussed. The increasing or reducing domination numbers using cardinality are also studied. Bondage $(\alpha(G))$ and non-bondage $(\alpha_K(G))$ sets are defined in regular intuitionistic fuzzy graph. The properties of bondage and non-bondage number of intuitionistic fuzzy graph analyzed. A minimum $2$-bondage set $X$ of an intuitionistic fuzzy graph (IFG) $G$ is a bondage set of regular intuitionistic fuzzy graph in $G$.

An Embedding-Driven Multi-Hop Spatio-Temporal Attention Network for Traffic Prediction

Traffic prediction is an important part of modern intelligent transportation systems (ITS), which helps transportation management and city planning. However, it is a very challenging task for modeling complex spatio-temporal dependencies, since the traffic data belongs to highly periodic multivariate time series which makes it hard to model accurate spatial dependencies only from time series and observed geolocation information of road segments. The existing research mainly focuses on finding ways of capturing dynamic spatial dependencies of road segments while neglecting the importance of periodicity, and few studies have explored a pure embedding-driven method that is robust to corrupted data to model periodicity. In this paper, we propose an embedding-driven multi-hop spatio-temporal attention network for traffic prediction (), which mainly focuses on leveraging the multi-scale periodicity of traffic data. Specifically, the proposed network applies a designed Fourier-series-based embedding, to capture the periodicity, which is more in line with real-world facts. Driven by the designed embedding, both local and global temporal dependencies are modeled properly by combining the attention-based methods and the convolution-based methods. Besides, we implement a trial that can hardly be seen in the existing traffic prediction works to combine the graph self-attention mechanism with a multi-hop diffusion process to explore the large-scale structural information on a designed set of graphs. Experiments on two real-world traffic datasets which contains traffic speed data for months show the effectiveness of our proposed methods. The experiments also suggest the methods can provide stable reasonable and smooth predictions for completely corrupted data.

Characterizations of $J$-Total Dominating Sets in Some Special Graphs and Graphs under Some Operations

Let G be a graph with no isolated vertex. A subset M ⊆ V (G) is called a J-open set if NG(a)\NG(b) ̸= ∅ and NG(b)\NG(a) ̸= ∅ ∀ a, b ∈ M, where a ̸= b. If in addition, M is a total dominating in G, then we call M a J-total dominating set in G. The maximum cardinality amongall J-total dominating set in G, denoted by γJt(G), is called the J-total domination number of G. In this paper, we characterize J-total dominating sets in some special graphs and join of two graphs, and we use these results to obtain formulas for the parameters of these graphs. Moreover, we determine its relationships with other known parameters in graph theory. Finally, we derive the lower bound of the parameter for the corona of two graphs.

Resolving Independent Dominating Set pada Graf Bunga, Graf Gear, dan Graf Bunga Matahari

Resolving independent dominating set is the development of metric dimension and independent dominating set. Resolving independent dominating sets is a concept which discusses about determining the minimum vertex on a graph provided that the vertex that becomes the dominating set can dominate the surrounding vertex and there are no two adjacent vertices dominator and also meet the requirement of metric dimension where each vertex in graph G must have a different representation which respect to the resolving independent dominating set . In this study, we examined the resolving independent dominating set of flower graphs, gear graphs, and sunflower graphs.

An algebraic approach to sets defining minimal dominating sets of regular graphs

An algebraic approach to sets defining minimal dominating sets of regular graphs

On the Monitoring-Edge-Geodetic Numbers of Line Graphs

For a vertex set [Formula: see text], we say that [Formula: see text] is a monitoring-edge-geodetic set (MEG-set for short) of graph [Formula: see text], that is, some vertices of [Formula: see text] can monitor an edge of the graph, if and only if we can remove that edge would change the distance between some pair of vertices in the set. The monitoring-edge-geodetic number [Formula: see text] of a graph [Formula: see text] is defined as the minimum cardinality of a monitoring-edge-geodetic set of [Formula: see text]. The line graph [Formula: see text] of [Formula: see text] is the graph whose vertices are in one-to-one correspondence with the edges of [Formula: see text], that is, if two vertices are adjacent in [Formula: see text] if and only if the corresponding edges have a common vertex in [Formula: see text]. In this paper, we study the relation between [Formula: see text] and [Formula: see text], and prove that [Formula: see text]. Next, we have determined the exact values for a MEG-set of some special graphs and their line graphs. For a graph [Formula: see text] and its line graph [Formula: see text], we prove that [Formula: see text] can be arbitrarily large.

Algorithms and hardness results for edge total domination problem in graphs

For a graph G=(V,E) without an isolated edge, a set D⊆E is an edge total dominating set of G if every edge e∈E is adjacent to at least one edge of D. The Minimum Edge Total Dominating Set problem is to compute a minimum cardinality edge total dominating set of G. It is known that the decision version of the problem is NP-complete for bipartite graphs, chordal graphs, and planar graphs. On the positive side, the problem is efficiently solvable for trees. In this paper, we further study the complexity of this problem in other graph classes. We design a linear-time algorithm to compute a minimum cardinality edge total dominating set in chain graphs, a subclass of bipartite graphs. We also propose linear-time algorithms for two subclasses of chordal graphs, namely, split graphs and proper interval graphs with no cut vertices. In addition, we show that the problem is APX-complete for graphs with maximum degree 3 and propose an approximation algorithm for the problem in k-regular graphs, where k≥4. We discuss the complexity difference between the Minimum Edge Dominating Set problem and Minimum Edge Total Dominating Set problem which seem to be closely related.

The extremal spectral radius of generalized block graphs

A block graph is a graph in which every block is a complete graph. Denote by Kn,q the set of block graphs with n vertices and all blocks on q+1 vertices for every q≥2. Recently, Zhao and Liu (2023) determined the minimum spectral radius of graphs in Kn,q, which verified a conjecture posed by Conde et al. (2022). Replacing the complete graph by a general block or a cycle, we define a generalized block graph or a cycle tree, respectively. Let Bn,q (resp. Cn,q) be the set of generalized block graphs (resp. cycle trees) on n vertices and in which each block is of order q+1 for q≥2. In this paper, we obtain the maximum/minimum spectral radius of a graph in Cn,q. Furthermore, we determine the extremal graph attaining the minimum spectral radius among graphs in Bn,q, which coincides with that in Cn,q.

INFORMATION TECHNOLOGY OF FORMING THE EDUCATIONAL NETWORK OF THE TERRITORIAL COMMUNITY

Context. Local government organizations have significant decision-making power in the field of education. This requires the development of understandable tools that help form a network of educational institutions that provide high-quality educational services. The object of the study was the process of forming an educational network (ENW) of the community in the implementation of territorial development projects as part of the reform of local self-government.
 Objective. The goal of the work is to increase objectivity of decisions made in the formation of the ENW territorial community, summarizing demographic, infrastructural, personnel, economic factors when choosing the location of ENW facilities.
 Method. The study of the classical placement problem and its further adaptation to real problems arising from the implementation of the education reform made it possible to represent ENW territorial communities as a set of independent complete bipartite graphs. In this case, to solve the problem of choosing the location of an educational institution on the network, an information technology (IT) was developed to form an ENW territorial community. Based on the adapted p-median model and methods of geospatial analysis, generalizing the requirements of the current legislation, a set of input and output parameters of IT and a set of its operations are formed. The representation of the IT structure in the form of an IDEF0-model clearly explains how a combination of various factors is processed and generalized when making decisions while creating ENW and looking for ways to improve it.
 Results. The developed IT was investigated in solving the problem of forming a network of lyceums in the Kharkiv region using geospatial information, open statistical data and data on lyceums. The proposed accommodation options make it possible to achieve a general level of accessibility of specialized secondary education of almost 94%. IT has additional tools for solving the problems of transporting students to the place of study and home.
 Conclusions. The experiments carried out confirmed the operability of the proposed IT. The generalization of the results obtained makes it possible to recommend it for practical use in solving the problems of analyzing the current state of ENW, finding ways to improve it and possible directions for development, as well as evaluating solutions related to the spatial planning of ENW.

Quantum algorithm for minimum dominating set problem with circuit design

Using quantum algorithms to solve various problems has attracted widespread attention with the development of quantum computing. Researchers are particularly interested in using the acceleration properties of quantum algorithms to solve NP-complete problems. This paper focuses on the well-known NP-complete problem of finding the minimum dominating set in undirected graphs. To expedite the search process, a quantum algorithm employing Grover’s search is proposed. However, a challenge arises from the unknown number of solutions for the minimum dominating set, rendering direct usage of original Grover’s search impossible. Thus, a swap test method is introduced to ascertain the number of iterations required. The oracle, diffusion operators, and swap test are designed with achievable quantum gates. The query complexity is O(1.414 n ) and the space complexity is O(n). To validate the proposed approach, qiskit software package is employed to simulate the quantum circuit, yielding the anticipated results.

Ad-Hoc Lanzhou Index

This paper initiates the study of the mathematical aspects of the ad-hoc Lanzhou index. If G is a graph with the vertex set {x1,…,xn}, then the ad-hoc Lanzhou index of G is defined by Lz˜(G)=∑i=1ndi(n−1−di)2, where di represents the degree of the vertex xi. Several identities for the ad-hoc Lanzhou index, involving some existing topological indices, are established. The problems of finding graphs with the extremum values of the ad-hoc Lanzhou index from the following sets of graphs are also attacked: (i) the set of all connected ξ-cyclic graphs of a fixed order, (ii) the set of all connected molecular ξ-cyclic graphs of a fixed order, (iii) the set of all graphs of a fixed order, and (iv) the set of all connected molecular graphs of a fixed order.

On the Holroyd-Talbot conjecture for sparse graphs

Given a graph G, let μ(G) denote the size of the smallest maximal independent set of G. A family of sets is called a star if some element is in every set of the family. A split vertex has degree at least 3. Holroyd and Talbot conjectured the following Erdős-Ko-Rado-type statement about intersecting families of independent sets of graphs: if 1≤r≤μ(G)/2 then there is an intersecting family of independent r-sets of maximum size that is a star. In this paper we prove similar statements for sparse graphs on n vertices: roughly, for graphs of bounded average degree with r≤O(n1/3), for graphs of bounded degree with r≤O(n1/2), and for trees having a bounded number of split vertices with r≤O(n1/2).

Excluding a planar matching minor in bipartite graphs

The notion of matching minors is a specialisation of minors fit for the study of graphs with perfect matchings. Matching minors have been used to give a structural description of bipartite graphs on which the number of perfect matchings can be computed efficiently, based on a result of Little, by McCuaig et al. in 1999.In this paper we generalise basic ideas from the graph minor series by Robertson and Seymour to the setting of bipartite graphs with perfect matchings. We introduce a version of Erdős-Pósa property for matching minors and find a direct link between this property and planarity. From this, it follows that a class of bipartite graphs with perfect matchings has bounded perfect matching width if and only if it excludes a planar matching minor. We also present algorithms for bipartite graphs of bounded perfect matching width for a matching version of the disjoint paths problem, matching minor containment, and for counting the number of perfect matchings. From our structural results, we obtain that recognising whether a bipartite graph G contains a fixed planar graph H as a matching minor, and that counting the number of perfect matchings of a bipartite graph that excludes a fixed planar graph as a matching minor are both polynomial time solvable.

An old problem of Erdős: A graph without two cycles of the same length

In 1975, P. Erd\H{o}s proposed the problem of determining the maximum number $f(n)$ of edges in a graph on $n$ vertices in which any two cycles are of different lengths. Let $f^{\ast}(n)$ be the maximum number of edges in a simple graph on $n$ vertices in which any two cycles are of different lengths. Let $M_n$ be the set of simple graphs on $n$ vertices in which any two cycles are of different lengths and with the edges of $f^{\ast}(n)$. Let $mc(n)$ be the maximum cycle length for all $G \in M_n$. In this paper, it is proved that for $n$ sufficiently large, $mc(n)\leq \frac{15}{16}n$. We make the following conjecture: $$\lim_{n \rightarrow \infty} {mc(n)\over n}= 0.$$

A fast and differentiable optimization model for finding the minimum dominating set in large-scale graphs

The minimum dominating set is an important NP-hard problem in graph theory, widely used in various fields such as computer network layout and social networks. Although linear programming algorithms can obtain a global optimal solution on small graphs, they are not suitable for large graphs. To find a local optimal solution, many heuristic algorithms have been proposed. These algorithms require strong techniques and programming, and most of them still need improvement in performance on large graphs. This paper presents a cooperative–competitive model that transforms the discrete problem into a continuously differentiable problem. By using the gradient descent method, this approach can handle all vertices simultaneously, and its time complexity is linear. Compared to existing search algorithms, this method is based on a straightforward principle and is fast. For small graphs, it can achieve performance almost comparable to linear programming methods. For large graphs, this method can explore more diverse local optimal solutions in the same amount of time, making it easier to obtain better performance. In this paper, a comparison and analysis of this gradient-based algorithm and graph theory-based search algorithms are conducted on the minimum dominating set, allowing researchers to have a clearer understanding of the advantages and disadvantages of search algorithms and gradient methods in complex problems. The referenced search algorithms may potentially improve the gradient descent algorithm, leading to better solutions in other fields. At the same time, gradient-based methods can be used to solve many similar NP-hard problems in graph theory.

Some snarks are worse than others

Many conjectures and open problems in graph theory can either be reduced to cubic graphs or are directly stated for them. Furthermore, it is known that for a lot of problems, a counterexample must be a snark, that is, a bridgeless cubic graph which is not 3-edge-colourable. In this paper we deal with the fact that the family of potential counterexamples to many interesting conjectures can be narrowed even further to the family S≥5 of bridgeless cubic graphs whose edge set cannot be covered with four perfect matchings. The Cycle Double Cover Conjecture, the Shortest Cycle Cover Conjecture and the Fan–Raspaud Conjecture are examples of statements for which S≥5 is crucial. In this paper we study parameters which have the potential to further refine S≥5 and thus enlarge the set of cubic graphs for which the mentioned conjectures can be verified. We show that S≥5 can be naturally decomposed into subsets with increasing complexity, thereby producing a natural scale for proving these conjectures. More precisely, we consider the following parameters and questions: given a bridgeless cubic graph, (i) how many perfect matchings need to be added, (ii) how many copies of the same perfect matching need to be added, and (iii) how many 2-factors need to be added so that the resulting regular graph is Class I? We present new results for these parameters and we also establish some strong relations between these problems and some long-standing conjectures.

Locating fair domination in graphs

Graphs considered here are simple, finite and undirected. A graph is denoted by [Formula: see text] and its vertex set by [Formula: see text] and edge set by [Formula: see text]. Many researchers were attracted by two concepts introduced in [P. J. Slater, Domination and location in acyclic graphs, Networks 17 (1987) 55–64; P. J. Slater, Dominating and reference sets in graphs, J. Math. Phys. Sci. 22 (1988) 445–455; Y. Caro, A. Hansberg and M. Henning, Fair domination in graphs, Discrete Math. 312 (2012) 2905–2914]. One is locating domination and the other is fair domination. A subset [Formula: see text] of [Formula: see text] is called a locating dominating set of [Formula: see text] if for any [Formula: see text] and both sets are non-empty. [Formula: see text] is called a fair dominating set of [Formula: see text] for any [Formula: see text]. In this paper, both properties are combined and locating fair domination is studied.

Feedback vertex set reconfiguration in planar graphs

We study the complexity of deciding whether for two given feedback vertex sets of a graph there is a step-by-step transformation between them, such that for each feedback vertex set in the transformation, the next one is obtained by exchanging a single vertex. We give a classification of the complexity of this question for planar graphs in terms of the maximum degree. We show that for planar graphs of maximum degree at most three the problem is tractable because there always exists a transformation, while it is PSPACE-complete when the maximum degree is at most four. The positive side of the classification extends to K3,3-minor-free graphs of maximum degree three. We then consider the Matroid Parity problem, which generalizes feedback vertex sets in graphs of maximum degree three as well as matchings and spanning trees in general graphs. Generalizing known results for the latter two we show that there always exists a transformation between any two non-maximum independent parity sets.

Split Geodetic Dominating Sets in Path Graphs

Let be the family of split geodetic dominating sets of the path graph . with cardinality i and let . Then the split geodetic polynomial of is defined as = , where is the split geodetic domination number of .In this paper we have determined the family of split geodetic dominating sets of the path graph with cardinality .Also , we have obtained the recursive formula to derive the split geodetic domination polynomials of paths and also obtain some properties of this polynomial.