Set Of Graphs Research Articles

Domination and Independent Domination in Extended Supergrid Graphs

Supergrid graphs are derived by computing stitch paths for computerized embroidery machines. In the past, we have studied the Hamiltonian-related properties of supergrid graphs and their subclasses of graphs. In this paper, we propose a generalized graph class for supergrid graphs called extended supergrid graphs. Extended supergrid graphs include grid graphs, supergrid graphs, diagonal supergrid graphs, and triangular supergrid graphs as subclasses of graphs. In this paper, we study the problems of domination and independent domination on extended supergrid graphs. A dominating set of a graph is the subset of vertices on it, such that every vertex of the graph is in this set or adjacent to at least a vertex of this set. If any two vertices in a dominating set are not adjacent, this is called an independent dominating set. Domination and independent domination problems find a dominating set and an independent dominating set with the least number of vertices on a graph, respectively. The domination and independent domination set problems on grid graphs are known to be NP-complete, meaning that these two problems on extended supergrid graphs are also NP-complete. However, the complexities of these two problems in other subclasses of graphs remain unknown. In this paper, we first prove that these two problems on diagonal supergrid graphs are NP-complete, then, by a simple extension, we prove that these two problems on supergrid graphs and triangular supergrid graphs are also NP-complete. In addition, these two problems on rectangular supergrid graphs are known to be linearly solvable; however, the complexities of these two problems on rectangular triangular-supergrid graphs remain unknown. This paper provides tight upper bounds on the sizes of the minimum dominating and independent dominating sets for rectangular triangular-supergrid graphs.

On disjoint maximum and maximal independent sets in graphs and inverse independence number

In this paper, we give a class of graphs that do not admit disjoint maximum and maximal independent (MMI) sets. The concept of inverse independence was introduced by Bhat and Bhat in [Inverse independence number of a graph, Int. J. Comput. Appl. 42(5) (2012) 9–13]. Let [Formula: see text] be a [Formula: see text]-set in [Formula: see text]. An independent set [Formula: see text] is called an inverse independent set with respect to [Formula: see text]. The inverse independence number [Formula: see text] is the size of the largest inverse independent set in [Formula: see text]. Bhat and Bhat gave few bounds on the independence number of a graph, we continue the study by giving some new bounds and exact value for particular classes of graphs: spider tree, the rooted product and Cartesian product of two particular graphs.

Network reconstruction problem for an epidemic reaction--diffusion system

Abstract We study the network reconstruction problem for an epidemic reaction–diffusion system. These systems are an extension of deterministic, compartmental models to a graph setting, where the reactions within the nodes are coupled by a diffusion dynamics. We study the influence of the diffusion rate and the network topology, on the reconstruction and prediction problems, both from a theoretical and experimental standpoint. Results first show that for almost every network, the reconstruction problem is well posed. Then, we show that the faster the diffusion dynamics, the harder the reconstruction, but that increasing the sampling rate may help in this respect. Second, we demonstrate that it is possible to classify symmetrical networks generating the same trajectories, and that the prediction problem can still be solved satisfyingly, even when the network topology makes exact reconstruction difficult.

Synthesis and structural studies on the supramolecular architecture of two novel proton transfer molecular salts of 2−aminopyridine

Herein, synthesis, analytical and structural characterization of two new molecular salts (AP−PHBA and AP−NA) of 2−aminopyridine (AP) using GRAS molecules i.e, p−hydroxybenzoic acid (PHBA) and nicotinic acid (NA) has been envisaged. The TGA/DSC revealed that both molecular salts were thermally more stable than the AP molecule. In the UV-Vis spectrum of AP−PHBA there observed a bathochromic shift (λmax 298 nm to 305 nm) caused by extended conjugation due to the hydroxy group at the para position of PHBA which might influence the photophysical property and no such shift was observed in AP−NA (λmax=298 nm). Both the salts were found to be dominated by the two−point charge assisted acid−pyridine heterosynthonic ring motif with graph set notation R22(8). This was held via N+−H···O− and N−H···O linkages that formed between the 2−aminopyridinium cation and carboxylate anions. In both crystals, a ribbon synthon linking the ring synthon via N2A—H2AB/AC···O3B hydrogen bonds was noted with the chain motifs C21(6) and C22(6). Furthermore, the C1A—H1AA···N1B and N2A—H2AC···O3B networks were associated to build a large tetramer with R44(22) was seen in the AP−NA salt system. From Hirshfeld surfaces analysis, it was noteworthy that the variation in the percentage contribution of O···H (20.8% in AP−PHBA and 15.8% in AP−NA) and N···H (1.9% in AP−PHBA and 12.1% in AP−NA) interconnects signify the importance of hydroxy group (PHBA) and pyridine N (NA) in stabilizing the crystal systems. Hence, a complete analysis of crystal structure and supramolecular packing of the two molecular salts AP−PHBA and AP−NA was presented.

Regular sets in Cayley graphs

In a graph $\Gamma$ with vertex set $V$, a subset $C$ of $V$ is called an $(a,b)$-perfect set if every vertex in $C$ has exactly $a$ neighbors in $C$ and every vertex in $V\setminus C$ has exactly $b$ neighbors in $C$, where $a$ and $b$ are nonnegative integers. In the literature $(0,1)$-perfect sets are known as perfect codes and $(1,1)$-perfect sets are known as total perfect codes. In this paper we prove that, for any finite group $G$, if a non-trivial normal subgroup $H$ of $G$ is a perfect code in some Cayley graph of $G$, then it is also an $(a,b)$-perfect set in some Cayley graph of $G$ for any pair of integers $a$ and $b$ with $0\leqslant a\leqslant|H|-1$ and $0\leqslant b\leqslant |H|$ such that $\gcd(2,|H|-1)$ divides $a$. A similar result involving total perfect codes is also proved in the paper.

Squared Energy-Resolved Mass Spectrometry Advances Quantitative Bile Acid Submetabolome Characterization.

The bile acid (BA) submetabolome can partially reflect either physiological or pathological status of vertebrates. The structural diversity, however, extensively hinders BA submetabolome clarification. Here, efforts were primarily devoted to enhance structural annotation confidences of BAs, in particular the conjugated BAs, through fortifying a new technology, namely, squared energy-resolved mass spectrometry (ER2-MS), to traditional liquid chromatography with tandem mass spectrometry (LC-MS/MS). Because of possessing two tandem-in-space collision cells, namely, q2 and linear ion trap (LIT) chambers, Qtrap-MS was employed as the fit-for-purpose tool to conduct ER2-MS measurements. The first ER-MS was undertaken in a q2 cell to gain first-generation breakdown graphs to disclose conjugation sites via applying the multiple-reaction monitoring (MRM) program, and the second ER-MS was accomplished in a LIT chamber through programming MRM cubed to acquire second-generation breakdown graphs of concerned ions for scaffold characterization. An authentic BA library consisting of commercial BAs together with their in vitro metabolites was built to record a reference breakdown graph set. Moreover, the so-called universal metabolome standard sample that was prepared by pooling diverse BA-enriched matrices was applied for structural deciphering potential evaluation and quasi-quantitative analysis of all detected BAs as well, according to applying a well-defined quasi-content concept. High-confidence structural analysis was achieved for as many as 201 BAs, and significant impacts occurred for the BA submetabolome of HepG2 cells after lithocholic acid treatment. Together, ER2-MS provides a promising tool to promote, although not limited to, LC-MS/MS-based BA-targeted metabolomics.

Feedback Vertex Sets in (Directed) Graphs of Bounded Degeneracy or Treewidth

We study the minimum size $f$ of a feedback vertex set in directed and undirected $n$-vertex graphs of given degeneracy or treewidth. In the undirected setting the bound $\frac{k-1}{k+1}n$ is known to be tight for graphs with bounded treewidth $k$ or bounded odd degeneracy $k$. We show that neither of the easy upper and lower bounds $\frac{k-1}{k+1}n$ and $\frac{k}{k+2}n$ can be exact for the case of even degeneracy. More precisely, for even degeneracy $k$ we prove that $f < \frac{k}{k+2}n$ and for every $\epsilon>0$, there exists a $k$-degenerate graph for which $f\geq \frac{3k-2}{3k+4}n -\epsilon$.
 For directed graphs of bounded degeneracy $k$, we prove that $f\leq\frac{k-1}{k+1}n$ and that this inequality is strict when $k$ is odd. For directed graphs of bounded treewidth $k\geq 2$, we show that $f \leq \frac{k}{k+3}n$ and for every $\epsilon>0$, there exists a $k$-degenerate graph for which $f\geq \frac{k-2\lfloor\log_2(k)\rfloor}{k+1}n -\epsilon$. Further, we provide several constructions of low degeneracy or treewidth and large $f$.

Parameterized Inapproximability of Independent Set in H-Free Graphs

We study the Independent Set problem in H-free graphs, i.e., graphs excluding some fixed graph H as an induced subgraph. We prove several inapproximability results both for polynomial-time and parameterized algorithms. Halldórsson [SODA 1995] showed that for every \(\delta >0\) the Independent Set problem has a polynomial-time \((\frac{d-1}{2}+\delta )\)-approximation algorithm in \(K_{1,d}\)-free graphs. We extend this result by showing that \(K_{a,b}\)-free graphs admit a polynomial-time\({\mathcal {O}}(\alpha (G)^{1-1/a})\)-approximation, where \(\alpha (G)\) is the size of a maximum independent set in G. Furthermore, we complement the result of Halldórsson by showing that for some \(\gamma =\Theta (d/\log d),\) there is no polynomial-time \(\gamma \)-approximation algorithm for these graphs, unless NP = ZPP. Bonnet et al. [Algorithmica 2020] showed that Independent Set parameterized by the size k of the independent set is W[1]-hard on graphs which do not contain (1) a cycle of constant length at least 4, (2) the star \(K_{1,4}\), and (3) any tree with two vertices of degree at least 3 at constant distance. We strengthen this result by proving three inapproximability results under different complexity assumptions for almost the same class of graphs (we weaken conditions (1) and (2) that G does not contain a cycle of constant length at least 5 or \(K_{1,5}\)). First, under the ETH, there is no \(f(k) \cdot n^{o(k/\log k)}\) algorithm for any computable function f. Then, under the deterministic Gap-ETH, there is a constant \(\delta >0\) such that no \(\delta \)-approximation can be computed in \(f(k) \cdot n^{O(1)}\) time. Also, under the stronger randomized Gap-ETH there is no such approximation algorithm with runtime \(f(k) \cdot n^{o(\sqrt{k})}\). Finally, we consider the parameterization by the excluded graph H, and show that under the ETH, Independent Set has no \(n^{o(\alpha (H))}\) algorithm in H-free graphs. Also, we prove that there is no \(d/k^{o(1)}\)-approximation algorithm for \(K_{1,d}\)-free graphs with runtime \(f(d,k) \cdot n^{{\mathcal {O}}(1)}\), under the deterministic Gap-ETH.

CHART OF STATUS OF THE UNMANNED AVIATION COMPLEX

A scheme of interrelationships of external and internal macro- and microenvironmental factors and their influence on the states of the unmanned aerial vehicle system has been developed. It is proposed to use an approach with elements of graph theory and set theory to analyze the states of an unmanned aircraft complex. The given graph of the functioning of the man-machine system of a sample of the unmanned aircraft complex of the “micro UAS unmanned aircraft system” class sets the transitions that determine the closed cycle of the execution of a combat mission.
 An example of building a structure of functional indicators of unmanned aerial vehicles with life cycle elements using graph theory for the analysis of “bottlenecks” in ensuring reliability is given. The analysis of “bottlenecks” in the control systems of unmanned aerial vehicles in the area of the ground control station – unmanned aerial vehicle is aimed at determining directions for increasing reliability, flight safety, increasing functionality, flight range and improving ergonomic characteristics.
 A comprehensive analysis of the functioning and dynamics of changes in reliability indicators with relevant data for previous periods of operation will allow us to draw specific conclusions about the actual level of reliability of the UAS un manned aircraft system fleet and identify bottlenecks for further improvement and the possibility of transitioning to the use of third-generation BpAK (autonomous devices) with self-learning systems with artificial intelligence.
 The theoretical rationale and the given practical recommendations should be taken into account when developing new prospective models of unmanned aerial vehicles for the Armed Forces of Ukraine in accordance with their official purpose.

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.

Labelled well-quasi-order for permutation classes

While the theory of labelled well-quasi-order has received significant attention in the graph setting, it has not yet been considered in the context of permutation patterns. We initiate this study here, and show how labelled well quasi order provides a lens through which to view and extend previous well-quasi-order results in the permutation patterns literature. Connections to the graph setting are emphasised throughout. In particular, we establish that a permutation class is labelled well-quasi-ordered if and only if its corresponding graph class is also labelled well-quasi-ordered.Mathematics Subject Classifications: 05A05, 06A07Keywords: Labelled well-quasi-order, permutation patterns, well-quasi-order

Faster Graph Coloring in Polynomial Space

We present a polynomial-space algorithm that computes the number of independent sets of any input graph in time O(1.1389^n) for graphs with maximum degree 3 and in time O(1.2356^n) for general graphs, where n is the number of vertices in the input graph. Together with the inclusion-exclusion approach of Björklund, Husfeldt, and Koivisto [SIAM J. Comput. 2009], this leads to a faster polynomial-space algorithm for the graph coloring problem with running time O(2.2356^n) as well as an exponential-space O(1.2330^n) time algorithm for counting independent sets. Our main algorithm counts independent sets in graphs with maximum degree at most 3 and no vertex with three neighbors of degree 3. This polynomial-space algorithm is designed and analyzed using the recently introduced Separate, Measure and Conquer approach [Gaspers & Sorkin, ICALP 2015]. Using Wahlström’s compound measure approach, this improvement in running time for small degree graphs is then bootstrapped to larger degrees, giving the improvement for general graphs. Combining both approaches leads to some inflexibility in choosing vertices to branch on for the small-degree cases, which we counter by structural graph properties.

On the Inverse Symmetric Division Deg Index of Unicyclic Graphs

The symmetric division deg (SDD) index is among the 148 discrete Adriatic indices that were developed about a decade ago. Motivated by the success of the SDD index, Ghorbani et al. in a recent paper proposed the inverse version of this index and called it the inverse symmetric division deg (ISDD) index. In the aforementioned paper, the graphs possessing the maximum and minimum ISDD index over the set of all tree graphs having the given order were found. The present paper addresses the problem of finding the graphs having the largest and smallest ISDD index from the set of all connected unicyclic graphs having the specified order.

Rapid capture trajectories from parabolic orbit with modulated radial thrust

The aim of this paper is to analyze the performance of a radially-accelerated spacecraft in a capture mission scenario, in which a space vehicle transfers from a parabolic approaching trajectory of assigned semilatus rectum to a target circular orbit around a generic celestial body. The radial propulsive acceleration provided by the spacecraft propulsion system can be modulated within a suitable given range, and the radial thrust can be either inward- or outward-directed. The transfer mission performance is analyzed in an optimal framework, by minimizing the total flight time with an indirect approach. In this context, the paper proposes a semi-analytical method to reduce the computation time required to solve the optimal control problem. Using a proper description of the spacecraft dynamics in a dimensionless form, the paper reports a set of graphs and tables that allow the designer to obtain a rapid approximation of the optimal mission performance with an accuracy consistent with that of a preliminary trajectory design. Finally, the proposed approach is used to describe the minimum time capture trajectories in a mission scenario towards Ceres.

Binary Equilibrium Optimization Algorithm for Computing Connected Domination Metric Dimension Problem

We consider, in this paper, the NP-hard problem of finding the minimum connected domination metric dimension of graphs. A vertex set B of a connected graph G = (V, E) resolves G if every vertex of G is uniquely identified by its vector of distances to the vertices in B. A resolving set B of G is connected if the subgraphB¯induced by B is a nontrivial connected subgraph of G. A resolving set is dominating if every vertex of G that does not belong to B is a neighbor to some vertices in B. The cardinality of the smallest resolving set of G, the cardinality of the minimal connected resolving set, and the cardinality of the minimal connected domination resolving set are the metric dimension of G, connected metric dimension of G, and connected domination metric dimension of G, respectively. We present the first attempt to compute heuristically the minimum connected dominant resolving set of graphs by a binary version of the equilibrium optimization algorithm (BEOA). The particles of BEOA are binary-encoded and used to represent which one of the vertices of the graph belongs to the connected domination resolving set. The feasibility is enforced by repairing particles such that an additional vertex generated from vertices of G is added to B, and this repairing process is iterated until B becomes the connected domination resolving set. The proposed BEOA is tested using graph results that are computed theoretically and compared to competitive algorithms. Computational results and their analysis show that BEOA outperforms the binary Grey Wolf Optimizer (BGWO), the binary Particle Swarm Optimizer (BPSO), the binary Whale Optimizer (BWO), the binary Slime Mould Optimizer (BSMO), the binary Grasshopper Optimizer (BGO), the binary Artificial Ecosystem Optimizer (BAEO), and the binary Elephant Herding Optimizer (BEHO).

Crystal structure of diclazuril, C17H9Cl3N4O2

The crystal structure of diclazuril has been solved and refined using synchrotron X-ray powder diffraction data, and optimized using density functional theory techniques. Diclazuril crystallizes in space group P21/a (#14) with a = 27.02080(18), b = 11.42308(8), c = 5.36978(5) Å, β = 91.7912(7)°, V = 1656.629(15) Å3, and Z = 4. The crystal structure consists of layers of molecules parallel to the ac-plane. A strong N–H⋯O hydrogen bond links the molecules into dimers along the a-axis with a graph set R2,2(8). The powder pattern has been submitted to ICDD for inclusion in the Powder Diffraction File™ (PDF®).

Crystal structure of ponazuril, C18H14F3N3O6S

The crystal structure of ponazuril has been solved and refined using synchrotron X-ray powder diffraction data, and optimized using density functional theory techniques. Ponazuril crystallizes in space group P21/c (#14) with a = 8.49511(6), b = 12.38696(6), c = 18.84239(17) Å, β = 96.7166(4)°, V = 1969.152(12) Å3, and Z = 4. N–H⋯O hydrogen bonds link the molecules into chains along the a-axis, with a graph set C1,1(6). The powder pattern has been submitted to ICDD for inclusion in the Powder Diffraction File™ (PDF®).

Bounds for the connected domination number of maximal outerplanar graphs

A dominating set of a graph G is a set S ⊆ V ( G ) such that every vertex in G is either in S or adjacent to a vertex in S . A dominating set S is a connected dominating set if the subgraph of G induced by S is connected. The minimum cardinality of a connected dominating set of G is the connected domination number , denoted by γ c ( G ) . Zhuang showed that γ c ( G ) of a maximal outerplanar graph G is bounded by min { ⌊ n + k 2 ⌋ − 2 , ⌊ 2 ( n − k ) 3 ⌋ } (Zhuang, 2020), where k is the number of 2-degree vertices in G . In this paper, we give an algorithm for finding a connected dominating set of maximal outerplanar graphs and get an upper bound γ c ( G ) ≤ ⌊ n − k + x 2 ⌋ , where x is a counter in the algorithm and x ≤ k − 2 . As a corollary, the result that γ c ( G ) ≤ ⌊ n − 2 2 ⌋ for a maximal outerplanar G is gotten directly. This results is better than the above known bound for 3 < k < n + 6 4 . In addition, we complement some analysis with simulations to evaluate the advantages of our results.

Evolving subgraph matching on temporal graphs

Given a set of query graphs and specified orders between these query graphs, the evolving subgraph matching problem finds all subgraph sets from the temporal graph that not only match the given query graph set but also satisfy the specified orders. This problem motivates many significant application, e.g., social network analysis, financial fraud crimes detection, biological information mining, and so on. To efficiently address this problem, an evolving subgraph matching algorithm based on two key techniques is proposed. First, a memory-efficient index structure called EMI is designed to compactly store all necessary information required for finding all evolving matchings. To make the EMI compact and to construct the EMI efficiently, several techniques, such as four well-designed filters and adaptive filling orders, are proposed. Second, an evolving subgraph matching enumeration method is designed to immediately run on the EMI instead of on the temporal graph. The enumeration method expands evolving matchings in an edge-by-edge manner. Since the EMI is compactly maintained in the main memory and generally smaller than the temporal graph, the enumeration method runs very fast on the EMI. An extensive experimental evaluation has been carried out to evaluate the proposed algorithm. The experimental results show that the enumeration method executed on the EMI is an order of magnitude faster and requires significantly less memory than the baseline algorithms. To the best of our knowledge, there is no prior work on the evolving subgraph matching on temporal graphs.

CAMIG: Concurrency-Aware Live Migration Management of Multiple Virtual Machines in SDN-Enabled Clouds

By integrating Software-Defined Networking and cloud computing, virtualized networking and computing resources can be dynamically reallocated through live migration of Virtual Machines (VMs). Dynamic resource management such as load balancing and energy-saving policies can request multiple migrations when the algorithms are triggered periodically. There exist notable research efforts in dynamic resource management that alleviate single migration overheads, such as single migration time and co-location interference while selecting the potential VMs and migration destinations. However, by neglecting the resource dependency among potential migration requests, the existing solutions of dynamic resource management can result in the Quality of Service (QoS) degradation and Service Level Agreement (SLA) violations during the migration schedule. Therefore, it is essential to integrate both single and multiple migration overheads into VM reallocation planning. In this paper, we propose a concurrency-aware multiple migration selector that operates based on the maximal cliques and independent sets of the resource dependency graph of multiple migration requests. Our proposed method can be integrated with existing dynamic resource management policies. The experimental results demonstrate that our solution efficiently minimizes migration interference and shortens the convergence time of reallocation by maximizing the multiple migration performance while achieving the objective of dynamic resource management.