Set Of Graphs Research Articles

On conflict-free cuts: Algorithms and complexity

One way to define the Matching Cut problem is: Given a graph G, is there an edge-cut M of G such that M is an independent set in the line graph of G? We propose the more general Conflict-Free Cut problem: Together with the graph G, we are given a so-called conflict graph Gˆ on the edges of G, and we ask for an edge-cutset M of G that is independent in Gˆ. Since conflict-free settings are popular generalizations of classical optimization problems and Conflict-Free Cut was not considered in the literature so far, we start the study of the problem. We show that the problem is NP-complete even when the maximum degree of G is 5 and Gˆ is 1-regular. The same reduction implies an exponential lower bound on the solvability based on the Exponential Time Hypothesis. We also give parameterized complexity results: We show that the problem is fixed-parameter tractable with the vertex cover number of G as a parameter, and we show W[1]-hardness even when G has a feedback vertex set of size one, and the clique cover number of Gˆ is the parameter. Since the clique cover number of Gˆ is an upper bound on the independence number of Gˆ and thus the solution size, this implies W[1]-hardness when parameterized by the cut size. We list polynomial-time solvable cases and interesting open problems. At last, we draw a connection to a symmetric variant of SAT.

Quantum-Informed Recursive Optimization Algorithms

We propose and implement a family of quantum-informed recursive optimization (QIRO) algorithms for combinatorial optimization problems. Our approach leverages quantum resources to obtain information that is used in problem-specific classical reduction steps that recursively simplify the problem. These reduction steps address the limitations of the quantum component (e.g., locality) and ensure solution feasibility in constrained optimization problems. Additionally, we use backtracking techniques to further improve the performance of the algorithm without increasing the requirements on the quantum hardware. We showcase the capabilities of our approach by informing QIRO with correlations from classical simulations of shallow circuits of the quantum approximate optimization algorithm, solving instances of maximum independent set and maximum satisfiability problems with hundreds of variables. We also demonstrate how QIRO can be deployed on a neutral atom quantum processor to find large independent sets of graphs. In summary, our scheme achieves results comparable to classical heuristics even with relatively weak quantum resources. Furthermore, enhancing the quality of these quantum resources improves the performance of the algorithms. Notably, the modular nature of QIRO offers various avenues for modifications, positioning our work as a template for a broader class of hybrid quantum-classical algorithms for combinatorial optimization. Published by the American Physical Society 2024

Graphs obtained by disjoint unions and joins of cliques and stable sets

We consider the set of graphs that can be constructed from a one-vertex graph by repeatedly adding a clique or a stable set linked to all or none of the vertices added in previous steps. This class of graphs contains various well-studied graph families such as threshold, domishold, co-domishold and complete multipartite graphs, as well as graphs with linear clique-width at most 2. We show that it can be characterized by three forbidden induced subgraphs as well as by properties involving maximal stable sets and minimal dominating sets. We also give a simple recognition algorithm and formulas for the computation of the stability and domination numbers of these graphs.

Structure-property modeling of coumarins and coumarin-related compounds in pharmacotherapy of cancer by employing graphical topological indices.

Coumarins, a subgroup of colorless and crystalline oxygenated heterocyclic compounds originally discovered in the plant Dipteryx odorata, were the subject of a recent study investigating their quantitative structure-activity relationship (QSAR) in cancer pharmacotherapy. This study utilized graph theoretical molecular descriptors, also known as topological indices, as a numerical representation method for the chemical structures embedded in molecular graphs. These descriptors, derived from molecular graphs, play a pivotal role in quantitative structure-property relationship (QSPR) analysis. In this paper, intercorrelation between the Balban index, connective eccentric index, eccentricity connectivity index, harmonic index, hyper Zagreb index, first path Zagreb index, second path Zagreb index, Randic index, sum connectivity index, graph energy and Laplacian energy is studied on the set of molecular graphs of coumarins. It is found that the pairs of degree-based indices are highly intercorrelated. The use of these molecular descriptors in structure-boiling point modeling was analyzed. Finally, the curve-linear regression between considered molecular descriptors with physicochemical properties of coumarins and coumarin-related compounds is obtained.

2-distance Zero Forcing Sets in Graphs

In this paper, we introduce new concept in graph theory called 2-distance zero forcing. We give some properties of this new parameter and investigate its connections with other parameters such as zero forcing and hop domination. We show that 2-distance zero forcing and hop domination (respectively, zero forcing parameter) are incomparable. Moreover, we characterize 2-distance zero forcing sets in some special graphs, and finally derive the exact values or bounds of the parameter using these results.

J-Open Independent Sets in Graphs

Let $G\neq \overline{K}_n$ be a graph with vertex and edge-sets $V(G)$ and $E(G)$, respectively. Then\linebreak $O$ $\subseteq$ $V(G)$ is called a J-open independent set of $G$ if for every $a,b \in V(G)$ where $a\neq b$, $d_G(a,b)$ $\neq 1$, and $N_G(a) \backslash N_G(b) \neq \varnothing$ and $N_G(b)\backslash N_G(a) \neq \varnothing$. The maximum cardinality of a J-open independent set of G, denoted by $\alpha_J(G)$, is called the J-open independence number of $G$. In this paper, we introduce new independence parameter called J-open independence. We show that this parameter is always less than or equal to the standard independence (resp. J-total domination) parameter of a graph. In fact, their differences can be made arbitrarily large. In addition, we show that J-open independence parameter is incomparable with hop independence parameter. Moreover, we derive some formulas and bounds of the parameter for some classes of graphs and the join of two graphs.

Formulas and Properties of 2-Hop Domination in Some Graphs

In this paper, 2-hop domination parameter is introduced and investigated on some special graphs and on the join of two graphs. Characterizations of 2-hop dominating sets in some special graphs are formulated to derive bounds or formulas of the parameter of these graphs. Moreover, new variant of pointwise non-domination is introduced to characterize 2-hop dominating sets in the join of two graphs. This characterization is used to calculate the exact value of 2-hop domination number of the join of two graphs.

A dynamic graph deep learning model with multivariate empirical mode decomposition for network‐wide metro passenger flow prediction

AbstractNetwork‐wide short‐term passenger flow prediction is critical for the operation and management of metro systems. However, it is challenging due to the inherent non‐stationarity, nonlinearity, and spatial–temporal dependencies within passenger flow. To tackle these challenges, this paper introduces a hybrid model called multi‐scale dynamic propagation spatial–temporal network (MSDPSTN). Specifically, the model employs multivariate empirical mode decomposition to jointly decompose the multivariate passenger flow into multi‐scale intrinsic mode functions. Then, a set of dynamic graphs is developed to reveal the passenger propagation law in metro networks. Based on the representation, a deep learning model is proposed to achieve multistep passenger flow prediction, which employs the dynamic propagation graph attention network with long short‐term memory to extract the spatial–temporal dependencies. Extensive experiments conducted on a real‐world dataset from Chengdu, China, validate the superiority of the proposed model. Compared to state‐of‐the‐art baselines, MSDPSTN reduces the mean absolute error, root mean squared error, and mean absolute percentage error by at least 3.243%, 4.451%, and 4.139%, respectively. Further quantitative analyses confirm the effectiveness of the components in MSDPSTN. This paper contributes to addressing inherent features of passenger flow to enhance prediction performance, offering critical insights for decision‐makers in implementing real‐time operational strategies.

Upper g- Eccentric Domination in Fuzzy Graphs

Objectives: To introduce a contemporary aspect notorious as upper g-eccentric point set, upper g-eccentric number, upper g-eccentric dominating set, upper g-eccentric domination number in fuzzy graphs and related concepts. Methods: Strong arc in fuzzy graphs concepts are used to find upper g-eccentric number and upper g-eccentric domination number. Findings: The new idea upper g-eccentric domination in fuzzy graphs will leads to numerous application such as cell phone tower formation in remote areas. Novelty: Using the concepts g-distance, geodesics and strong arc in fuzzy graphs, the upper g-eccentric dominating set and upper g-eccentric domination number in fuzzy graphs are obtained. Some points are observed in this concept and some theorem related to the upper g-eccentric dominant set, its numbers are proved. Keywords: g-Eccentric Dominating Set; g-Eccentric Domination Number; Upper g-Eccentric Domination Number

Local Multiset Dimension of Amalgamation Graphs.

Background: One of the topics of distance in graphs is the resolving set problem. Suppose the set W = { s 1, s 2, …, s k} ⊂ V ( G), the vertex representations of ∈ V ( G) is r m( x| W) = { d( x, s 1), d( x, s 2), …, d( x, s k)}, where d( x, s i) is the length of the shortest path of the vertex x and the vertex in W together with their multiplicity. The set W is called a local m-resolving set of graphs G if r m( v| W)≠ r m( u| W) for uv ∈ E( G). The local m-resolving set having minimum cardinality is called the local multiset basis and its cardinality is called the local multiset dimension of G, denoted by md l( G). Thus, if G has an infinite local multiset dimension and then we write . Methods: This research is pure research with exploration design. There are several stages in this research, namely we choose the special graph which is operated by amalgamation and the set of vertices and edges of amalgamation of graphs; determine the set W ⊂ V ( G); determine the vertex representation of two adjacent vertices in G; and prove the theorem. Results: The results of this research are an upper bound of local multiset dimension of the amalgamation of graphs namely md l( Amal( G, v, m)) ≤ m. md l( G) and their exact value of local multiset dimension of some families of graphs namely md l( Amal( P n, v, m)) = 1, , md l( Amal( W n, v, m)) = m. md l( W n), md l( Amal( F n, v, m)) = m. md l( F n) for d( v) = n, . Conclusions: We have found the upper bound of a local multiset dimension. There are some graphs which attain the upper bound of local multiset dimension namely wheel graphs.

Binary rat swarm optimizer algorithm for computing independent domination metric dimension problem

In this article, we look at the NP-hard problem of determining the minimum independent 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 recognized by its vector of distances to the vertices in B. If there are no neighboring vertices in a resolving set B of G, then B is independent. Every vertex of G that does not belong to B must be a neighbor of at least one vertex in B for a resolving set to be dominant. The metric dimension of G, independent metric dimension of G, and independent dominant metric dimension of G are, respectively, the cardinality of the smallest resolving set of G, the minimal independent resolving set, and the minimal independent domination resolving set. We propose the first attempt to use a binary version of the Rat Swarm Optimizer Algorithm (BRSOA) to heuristically calculate the smallest independent dominant resolving set of graphs. The search agent of BRSOA are binary-encoded and used to identify which one of the vertices of the graph belongs to the independent domination resolving set. The feasibility is enforced by repairing search agent such that an additional vertex created from vertices of G is added to B, and this repairing process is repeated until B becomes the independent domination resolving set. Using theoretically computed graph findings and comparisons to competing methods, the proposed BRSOA is put to the test. BRSOA surpasses the binary Grey Wolf Optimizer (BGWO), the binary Particle Swarm Optimizer (BPSO), the binary Whale Optimizer (BWOA), the binary Gravitational Search Algorithm (BGSA), and the binary Moth-Flame Optimization (BMFO), according to computational results and their analysis.

Reciprocal eigenvalue properties using the zeta and Möbius functions

In this paper, we develop a new approach to study the spectral properties of Boolean graphs using the zeta and Möbius functions on the Boolean algebra Bn of order 2n. This approach yields new proofs of the previously known results about the reciprocal eigenvalue property of Boolean graphs. Further, this approach allows us to extend the results to a more general setting of the zero-divisor graphs Γ(P) of complement-closed and convex subposets P of Bn. To do this, we consider the left linear representation of the incidence algebra of a poset P on the vector space of all real-valued functions V(P) on P. We then write down the adjacency operator A of the graph Γ(P) as the composition of two linear operators on V(P), namely, the operator that multiplies elements of V(P) on the left by the zeta function ζ of P and the complementation operator. This allows us to obtain the determinant of A and the inverse of A in terms of the Möbius function μ of the complement-closed posets P. Additionally, if we impose convexity on the poset P, then we obtain the strong reciprocal or strong anti-reciprocal eigenvalue property of Γ(P) and also obtain the absolute palindromicity of the characteristic polynomial of A. This produces a large family of examples of graphs having the strong reciprocal or strong anti-reciprocal eigenvalue property.

Two novel temperature-based topological indices with strong potential to predict physicochemical properties of polycyclic aromatic hydrocarbons with applications to silicon carbide nanotubes

For a vertex x ∈ V G , the temperature T x of x is defined as T x = d x /n − d x . A topological/graphical index G I is a map GI:∑→R , where ∑ (resp. R ) is the set of chemical graphs (resp. real numbers). Graphical indices are employed in structure-property & structure-activity modeling to predict physicochemical/thermodynamic/bilogical characteristics of a compound. A temperature-based graphical index of a chemical graph G is defined as GI T ≔ ∑edges f(T x , t y ), where f(T x , T y ) is a symmetric 2-variable map. In this paper, we introduce two new novel temperature-based indices known as the reduced reciprocal product-connectivity temperature (RRPT) index and the geometric-arithmetic temperature (GAT) index. The predictive potential of these indices have been investigated by employing them in structure-property modeling of physicochemical properties of polycyclic aromatic hydrocarbons (PAHs). The normal boiling point (bp) and the standard enthalpy of formation ΔHfo are selected as representatives of physicochemical characteristics. Intermolecular & van der Waals kind of interactions have been represented by bp, whereas, ΔHfo advocates for thermal characteristics of a compound. In order to validate the statistical inference, the lower 22 PAHs have been opted as test molecules as their experimental data is also publicly available. We propose a computational method to compute all temperature indices in literature and employ it to compute them for the lower 22 PAHs. Besides all the existing temperature indices, both RRPT & GAT are used in a quality testing to predict bp and ΔHfo for lower PAHs. Our statistical analysis asserts that both RRPT & GAT outperformed all the existing temperature indices for correlating bp and ΔHfo for lower PAHs. Most appropriate data-fitting regression models have been suggested to be linear. Since RRPT has the both of correlation coefficients >0.95, the study implicates its further employability in structure-property modeling. Importantly, our research contributes towards countering proliferation of graphical indices. Applications to well-performing temperature indices to correlate physicochemical characteristics of silicon carbide nanotubes are presented.

Maker-Breaker domination number for Cartesian products of path graphs $P_2$ and $P_n$

We study the Maker-Breaker domination game played by Dominator and Staller on the vertex set of a given graph. Dominator wins when the vertices he has claimed form a dominating set of the graph. Staller wins if she makes it impossible for Dominator to win, or equivalently, she is able to claim some vertex and all its neighbours. Maker-Breaker domination number $\gamma_{MB}(G)$ ($\gamma '_{MB}(G)$) of a graph $G$ is defined to be the minimum number of moves for Dominator to guarantee his winning when he plays first (second). We investigate these two invariants for the Cartesian product of any two graphs. We obtain upper bounds for the Maker-Breaker domination number of the Cartesian product of two arbitrary graphs. Also, we give upper bounds for the Maker-Breaker domination number of the Cartesian product of the complete graph with two vertices and an arbitrary graph. Most importantly, we prove that $\gamma'_{MB}(P_2\square P_n)=n$ for $n\geq 1$, $\gamma_{MB}(P_2\square P_n)$ equals $n$, $n-1$, $n-2$, for $1\leq n\leq 4$, $5\leq n\leq 12$, and $n\geq 13$, respectively. For the disjoint union of $P_2\square P_n$s, we show that $\gamma_{MB}'(\dot\cup_{i=1}^k(P_2\square P_n)_i)=k\cdot n$ ($n\geq 1$), and that $\gamma_{MB}(\dot\cup_{i=1}^k(P_2\square P_n)_i)$ equals $k\cdot n$, $k\cdot n-1$, $k\cdot n-2$ for $1\leq n\leq 4$, $5\leq n\leq 12$, and $n\geq 13$, respectively.

Transference for loose Hamilton cycles in random 3‐uniform hypergraphs

AbstractA loose Hamilton cycle in a hypergraph is a cyclic sequence of edges covering all vertices in which only every two consecutive edges intersect and do so in exactly one vertex. With Dirac's theorem in mind, it is natural to ask what minimum ‐degree condition guarantees the existence of a loose Hamilton cycle in a ‐uniform hypergraph. For and each , the necessary and sufficient such condition is known precisely. We show that these results adhere to a ‘transference principle’ to their sparse random analogues. The proof combines several ideas from the graph setting and relies on the absorbing method. In particular, we employ a novel approach of Kwan and Ferber for finding absorbers in subgraphs of sparse hypergraphs via a contraction procedure. In the case of , our findings are asymptotically optimal.

Split Detour Monophonic Sets in Graph

A subset T ⊆ V is a detourmonophonic set of G if each node (vertex) x in G contained in an p-q detourmonophonic path where p, q ∈ T.. The number of points in a minimum detourmonophonic set of G is called as the detourmonophonic number of G, dm(G). A subset T ⊆ V of a connected graph G is said to be a split detourmonophonic set of G if the set T of vertices is either T = V or T is detoumonophonic set and V – T induces a subgraph in which is disconnected. The minimum split detourmonophonic set is split detourmonophonic set with minimum cardinality and it is called a split detourmonophonic number, denoted by dms(G). For certain standard graphs, defined new parameter was identified. Some of the realization results on defined new parameters were established.

Structural Analysis of 3,5-Bistrifluoromethylhydrocinnamic Acid

The crystal structure of 3,5-bistrifluoromethylhydrocinnamic acid [systematic name: 3-[3,5-bis(trifluoromethyl)phenyl]propanoic acid], C11H8F6O2, has been determined and described. The structure was subject to the Hirshfeld surface-analysis and CE-B3LYP interaction-energies calculations. The title compound crystallises in the monoclinic P21/c space group with one molecule in the asymmetric unit. The propanoic acid side chain of the studied molecule has a bent conformation. The key supramolecular motif in the crystal structure is a centrosymmetric O–H∙∙∙O hydrogen-bonded dimer (R22(8) in the graph set notation). According to CE-B3LYP, the molecules involved in this motif exhibit the strongest pairwise interaction total energy (Etot = −67.9 kJ/mol). On the other hand, there are seven other interacting molecular pairs with significant Etot values in the range of −17 to −28 kJ/mol. In these, the energy is dominated by the dispersive contribution. A survey of the Cambridge Structural Database revealed that in other 3-phenylpropanoic acid structures, the middle dihedral angle of the propanoic acid side chain is always in the trans conformation. This contrasts the current structure where this dihedral angle is in the gauche conformation. According to the Density Functional Theory calculations in the gas phase (at the B3LYP/aug-cc-pvDZ level), the presence of the two CF3 groups (strong electron-withdrawing character) increases the population of the gauche conformers by a substituent electronic effect, and this may be a minor factor contributing to the appearance of this conformation observed in the solid state.

On linear spaces of bipartite graphs

The article considers symmetric linear spaces of bipartite graphs (SLSBG), i.e. the set of bipartite graphs with fixed lobes closed with respect to the symmetric difference and permutations of vertices in each lobe. The operation of symmetric difference itself is introduced in this work. The paper provides a structural description of all SLSBG. Symmetric linear spaces of bipartite graphs are divided into trivial (four SLSBG) and nontrivial. Non-trivial ones, in turn, are divided into two families. The first is C -series consisting only of bicomplete graphs, i.e. graphs that are a disjunct union of two complete bipartite graphs graph wings). The second family is D -series that includes graphs in which the degrees of vertices in one lobe have the same parity, and in the other lobe these degrees may be arbitrary. It is proved that every SLSBG of the D -series coincides with one of nine sets defined by the parity of the vertices’ degrees. For the SLSBG of the C -series it is obtained that every two-sided SLSBG (i.e., containing graphs whose both wings have nonempty lobes) is the intersection of the set of all bicomplete graphs with the set of all graphs with an even number of edges or with any space of the D -series.

Random Sampling Method of Large-Scale Graph Data Classification

Graph data appears in broad real-world applications in modelling complex objects in big data. Effective analysis of graph data provides a deeper understanding of the data in data mining tasks, including classification, clustering, prediction, and recommendation systems. Mining a large number of graphs becomes a challenging task because state-of-the-art methods are not scalable due to the memory limit. To address this issue, we propose a novel approximate random sampling method for large-scale graph data classification. In this approach, we applied a representation method to encode each graph as a record of a vector string and a set of graphs as a set of N records in a file. Then, we partition the set of records into disjoint subsets of data blocks, making each data block a random sample of the data file. After that, we randomly select a subset of data blocks, each being a random sample of the graph dataset, and compute the different graph property distributions. Since the data blocks in this model are much smaller than the entire data set, it is more efficient to analyze them on a standalone small machine, and multiple data blocks can be analyzed on multiple nodes of the cluster in parallel. Finally, we classified the graphs of data blocks using the SVM algorithm. In experimental evaluation, our proposed method outperformed state-of-the-art graph kernels on graph classification datasets in terms of accuracy.

Sparse graphs with bounded induced cycle packing number have logarithmic treewidth

A graph is Ok-free if it does not contain k pairwise vertex-disjoint and non-adjacent cycles. We prove that “sparse” (here, not containing large complete bipartite graphs as subgraphs) Ok-free graphs have treewidth (even, feedback vertex set number) at most logarithmic in the number of vertices. This is optimal, as there is an infinite family of O2-free graphs without K2,3 as a subgraph and whose treewidth is (at least) logarithmic.Using our result, we show that Maximum Independent Set and 3-Coloring in Ok-free graphs can be solved in quasi-polynomial time. Other consequences include that most of the central NP-complete problems (such as Maximum Independent Set, Minimum Vertex Cover, Minimum Dominating Set, Minimum Coloring) can be solved in polynomial time in sparse Ok-free graphs, and that deciding the Ok-freeness of sparse graphs is polynomial time solvable.