Intersection Graph Research Articles

Local distance antimagic cromatic number of join product of graphs with cycles or paths

Let $G$ be a graph of order $p$ without isolated vertices. A bijection $f: V \to \{1,2,3,\dots,p\}$ is called a local distance antimagic labeling, if $w_f(u)\ne w_f(v)$ for every edge $uv$ of $G$, where $w_f(u)=\sum_{x\epsilon N(u)} {f(x)}$. The local distance antimagic chromatic number $\chi_{lda}(G)$ is defined to be the minimum number of colors taken over all colorings of $G$ induced by local distance antimagic labelings of $G$. In this paper, we determined the local distance antimagic chromatic number of some cycles, paths, disjoint union of 3-paths. We also determined the local distance antimagic chromatic number of join products of some graphs with cycles or paths.

Bayesian reconstruction of Cartesian product graph signals with general patterns of missing data

In this paper, we address the challenge of signal reconstruction on Cartesian product graphs, which frequently arises in applications where node-level data may be corrupted by equipment failure or difficult to reliably collect. A recent approach to Graph Signal Reconstruction (GSR) is to formulate it as a Bayesian inference problem, where the underlying signal is presumed to be smooth with respect to the graph topology. However, the resulting linear system derived from this model specification is often intractable to solve directly for general Cartesian product graphs. To overcome this limitation, we introduce two robust and efficient iterative procedures for computing the posterior mean. The first method is a stationary iterative technique based on matrix splitting, while the second employs a conjugate gradient approach with a symmetric graph-spectral preconditioner. We provide a theoretical and numerical comparison of the convergence behaviour of these algorithms, demonstrating that the optimal choice depends on the model hyperparameters. Finally, we propose a novel supervised learning-based technique, enhanced by an active learning strategy, for estimating the posterior marginal variance. This circumvents the need to instantiate, invert or decompose large matrices directly in memory and results in an R2 statistic of over 0.95 on a synthetic test data set. This paper contributes to the development of advanced signal reconstruction techniques applicable in various domains, including large-scale sensor networks and social network analysis.

General fractal dimensions of graphs of products and sums of continuous functions and their decompositions

This study takes a broad approach to the fractal geometry problem and proposes an intrinsic definition of the general box dimensions and the general Hausdorff and packing dimensions by taking into account sums of the type ∑ih−1(sg(ri)) for some prescribed functions h, g and for all positive real s. Our primary aim is to conduct a more comprehensive exploration of the fractal dimensions exhibited by graphs resulting from the products and sums of two continuous functions. This article is entirely devoted to the investigation of this specific problem domain. In the course of our research, we furnish explicit formulas for computing both the general upper box dimension and the general lower box dimension associated with the graphs formed by the products and sums of two continuous functions. Additionally, we shed light on the decomposition of continuous functions into the sum of two continuous functions, employing the framework of the general lower and upper box dimensions. Furthermore, we establish an upper bound for the general upper box dimension applicable to each element within a finite ring of polynomials derived from continuous functions defined over the real number field R. Our inquiry also extends to the examination of general Hausdorff and packing dimensions for the graphs generated through both the summation and multiplication of two continuous functions.

Disjoint Perfect Secure Domination in the Cartesian Product and Lexicographic Product of Graphs

Let G be a graph. A dominating set D⊆V(G) is called a secure dominating set of G if for each vertex u∈V(G)∖D, there exists a vertex v∈D such that uv∈ E(G) and the set (D∖{v})∪{u} is a dominating set of G. If every u∈V(G)∖D is adjacent to exactly one vertex in D, then D is a perfect secure dominating set of G. Let D be a minimum perfect secure dominating set of G. If S⊆V(G)∖D is a perfect secure dominating set of G, then S is called an inverse perfect secure dominating set of G with respect to D. A disjoint perfect secure dominating set of G is the set C=D∪S⊆V(G). Furthermore, the disjoint perfect secure domination number, denoted by γ_ps γ_ps (G), is the minimum cardinality of a disjoint perfect secure dominating set of G. A disjoint perfect secure dominating set of cardinality γ_ps γ_ps (G) is called γ_ps γ_ps-set. In this paper, we extended the study on the concept of disjoint perfect secure domination in graphs. Furthermore, we characterized the disjoint perfect secure domination in the Cartesian product and lexicographic product of two graphs.

Graph Bipartization and Via Minimization for Intersection Graphs

The graph bipartization problem, arising from via minimization in VLSI design and related areas, consists in finding a vertex subset [Formula: see text] of graph [Formula: see text] such that the induced subgraph [Formula: see text] is bipartite and [Formula: see text] is maximized. The problem has been proved to be NP-hard even for planar graphs and cubic graphs. On the other hand, the study of polynomial-time algorithms for typical graph classes is significant in both theoretical and applied aspects. This paper focuses on several intersection graph classes, such as line graphs, circular-arc graphs, and directed path graphs. For the line graphs, we show the NP-hardness results in general and present the polynomial-time algorithms for special cases. For circular-arc graphs and directed path graphs, we propose algorithms that improve on the previously known ones.

Isoperimetric Inequalities and Supercritical Percolation on High-Dimensional Graphs

It is known that many different types of finite random subgraph models undergo quantitatively similar phase transitions around their percolation thresholds, and the proofs of these results rely on isoperimetric properties of the underlying host graph. Recently, the authors showed that such a phase transition occurs in a large class of regular high-dimensional product graphs, generalising a classic result for the hypercube. In this paper we give new isoperimetric inequalities for such regular high-dimensional product graphs, which generalise the well-known isoperimetric inequality of Harper for the hypercube, and are asymptotically sharp for a wide range of set sizes. We then use these isoperimetric properties to investigate the structure of the giant component L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1$$\\end{document} in supercritical percolation on these product graphs, that is, when p=1+ϵd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=\\frac{1+\\epsilon }{d}$$\\end{document}, where d is the degree of the product graph and ϵ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon >0$$\\end{document} is a small enough constant. We show that typically L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1$$\\end{document} has edge-expansion Ω1dlnd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega \\left( \\frac{1}{d\\ln d}\\right) $$\\end{document}. Furthermore, we show that L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1$$\\end{document} likely contains a linear-sized subgraph with vertex-expansion Ω1dlnd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega \\left( \\frac{1}{d\\ln d}\\right) $$\\end{document}. These results are best possible up to the logarithmic factor in d. Using these likely expansion properties, we determine, up to small polylogarithmic factors in d, the likely diameter of L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1$$\\end{document} as well as the typical mixing time of a lazy random walk on L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1$$\\end{document}. Furthermore, we show the likely existence of a cycle of length Ωndlnd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega \\left( \\frac{n}{d\\ln d}\\right) $$\\end{document}. These results not only generalise, but also improve substantially upon the known bounds in the case of the hypercube, where in particular the likely diameter and typical mixing time of L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1$$\\end{document} were previously only known to be polynomial in d.

Injective Coloring of Product Graphs

Injective Coloring of Product Graphs

A Note on Acyclic Coloring of Strong Product of Graphs

A Note on Acyclic Coloring of Strong Product of Graphs

Detecting k-Vertex Cuts in Sparse Networks via a Fast Local Search Approach

The <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> -vertex cut ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> -VC) problem belongs to the family of the critical node detection problems, which aims to find a minimum subset of vertices whose removal decomposes a graph into at least <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> connected components. It is an important NP-hard problem with various real-world applications, e.g., vulnerability assessment, carbon emissions tracking, epidemic control, drug design, emergency response, network security, and social network analysis. In this article, we propose a fast local search (FLS) approach to solve it. It integrates a two-stage vertex exchange strategy based on neighborhood decomposition and cut vertex, and iteratively executes operations of addition and removal during the search. Extensive experiments on both intersection graphs of linear systems and coloring/DIMACS graphs are conducted to evaluate its performance. Empirical results show that it significantly outperforms the state-of-the-art (SOTA) algorithms in terms of both solution quality and computation time in most of the instances. To evaluate its generalization ability, we simply extend it to solve the weighted version of the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> -VC problem. FLS also demonstrates its excellent performance.

Augmented Zagreb Index of Corona Product of Some Special Graphs

Abstract: Let GV, E be a simple undirected graph. The Augmented Zagreb Index of a graph G is defined as     3 2            uv E G du dv du dv AZI G , Where du is the degree of the vertex u in G . In this paper, the exact expression for Augmented Zagreb Index of different product of graphs like Comb, Wheel, Fan,andSun graph.

On the Crossing Numbers of Cartesian Products of Small Graphs with Paths, Cycles and Stars

There has been significant research dedicated towards computing the crossing numbers of families of graphs resulting from the Cartesian products of small graphs with arbitrarily large paths, cycles and stars. For graphs with four or fewer vertices, these have all been computed, but there are still various gaps for graphs with five or more vertices. We contribute to this field by determining the crossing numbers for fifteen such families.

The Restricted Edge-Connectivity of Strong Product Graphs

The restricted edge-connectivity of a connected graph G, denoted by λ′(G), if it exists, is the minimum cardinality of a set of edges whose deletion makes G disconnected, and each component has at least two vertices. It was proved that λ′(G) exists if and only if G has at least four vertices and G is not a star. In this case, a graph G is called maximally restricted edge-connected if λ′(G)=ξ(G), and a graph G is called super restricted edge-connected if each minimum restricted edge-cut isolates an edge of G. The strong product of graphs G and H, denoted by G⊠H, is the graph with the vertex set V(G)×V(H) and the edge set {(x1,y1)(x2,y2)|x1=x2 and y1y2∈E(H); or y1=y2 and x1x2∈E(G); or x1x2∈E(G) and y1y2∈E(H)}. In this paper, we determine, for any nontrivial connected graph G, the restricted edge-connectivity of G⊠Pn, G⊠Cn and G⊠Kn, where Pn, Cn and Kn are the path, cycle and complete graph of order n, respectively. As corollaries, we give sufficient conditions for these strong product graphs G⊠Pn, G⊠Cn and G⊠Kn to be maximally restricted edge-connected and super restricted edge-connected.

Formulation, Proposition and Application of Interval Semigraph

Semigraph represents a versatile extension within graph theory, offering a broader framework for exploration. Among these, intersection graphs hold pivotal roles, serving as essential analytical tools with wide-ranging applications. In this study, we introduce the integration of intersection graphs into the generalized structure of semigraphs, aiming to invigorate researchers’ interest and encourage further advancements in this domain. This paper introduces two novel concepts: edge intersection semigraphs and interval semigraphs. Additionally, foundational results pertaining to these concepts are established, elucidating their fundamental properties. Furthermore, we illustrate the practical utility of interval semigraphs through an illustrative example involving their application in optimizing road traffic management systems.

Improving the Principles of Identifying Critical Requirements for the Assembly of High-Precision Products

Introduction. The problem of improving the manufacturing of high-precision products is currently becoming a key one, since the requirements for them are constantly being tightened. Maintaining assembly quality and accuracy is an important aspect of manufacturing precision products. Standard approaches to this process do not always have sufficient versatility. Existing studies that aim to develop universal approaches, such as end-to-end production design or the application of parallel engineering principles, also have a number of shortcomings. These include the fact that the given approaches do not fully take into account information about the technology capabilities of a particular production when making design decisions, and do not consider the relationship between the manufacturing preparation of machining and mechanical assembly industries. That is why studies aimed at developing such universal approaches have high applicability. To solve these problems, the authors conceptually developed a set of formalized design procedures for a system of accounting requirements for the assembly of high-precision products in the design of machining processes. However, to effectively identify the numerous requirements for the assembly of high-precision products (output parameters) and select those that cannot be provided by the method of complete interchangeability (critical elements), additional research is needed. The research objective is to develop principles for identifying the output parameters of high-precision products and detecting critical elements. To achieve this goal, it is required to solve the following problems: to formulate principles for constructing generalized surface graphs of high-precision products; to develop standards for classifying output parameters and identifying critical ones.Materials and Methods. To conduct the research, a high-precision assembly unit was selected — “Stator Package 2”. The research was carried out under real conditions of the existing multiproduct manufacture. For this assembly, a generalized surface graph was constructed, including information about the nature and sequence of surfaces, requirements for the assembly, dimensional tolerances and tolerances of shape and location, with its subsequent analysis.Results. This paper presents the results of research on improving the enlarged block of design procedures for analyzing requirements for the assembly of high-precision products of the designed system. The paper established the relationship between the accuracy of dimensional tolerances and the tolerances of the shape and location of the surfaces of the product to which these dimensions belonged. Based on the relationship obtained, an order was determined for the unambiguous identification of critical elements.Discussion and Conclusion. The application of this technique makes it possible to increase the reliability of the source information obtained during the implementation of an enlarged block of design procedures, as well as the validity and efficiency of identifying rational manufacturing technologies at subsequent stages of the system implementation, while providing the specified quality, accuracy of products, and reducing the complexity and cost of their manufacture.

Equitable colorings of ��-corona products of cubic graphs

A graph G is equitably k-colorable if its vertices can be partitioned into k independent sets in such a way that the number of vertices in any two sets differ by at most one. The smallest integer k for which such a coloring exists is known as the equitable chromatic number of G and it is denoted by x=( G). In this paper the problem of determining the value of equitable chromatic number for multicoronas of cubic graphs G◦ lH is studied. The problem of ordinary coloring of multicoronas of cubic graphs is solvable in polynomial time. The complexity of equitable coloring problem is an open question for these graphs. We provide some polynomially solvable cases of cubical multicoronas and give simple linear time algorithms for equitable coloring of such graphs which use at most x=( G◦ lH) + 1 colors in the remaining cases.

Extending Partial Representations of Circle Graphs in Near-Linear Time

The partial representation extension problem generalizes the recognition problem for geometric intersection graphs. The input consists of a graph G, a subgraph H⊆G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H \\subseteq G$$\\end{document} and a representation R′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal R'$$\\end{document} of H. The question is whether G admits a representation R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal R$$\\end{document} whose restriction to H is R′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal R'$$\\end{document}. We study this question for circle graphs, which are intersection graphs of chords of a circle. Their representations are called chord diagrams. We show that for a graph with n vertices and m edges the partial representation extension problem can be solved in O((n+m)α(n+m))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O((n + m) \\alpha (n + m))$$\\end{document} time, thereby improving over an O(n3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(n^3)$$\\end{document}-time algorithm by Chaplick et al. (J Graph Theory 91(4), 365–394, 2019). The main technical contributions are a canonical way of orienting chord diagrams and a novel compact representation of the set of all canonically oriented chord diagrams that represent a given circle graph G, which is of independent interest.

Graph to Activation Energy Models Easily Reach Irreducible Errors but Show Limited Transferability.

Activation energy characterization of competing reactions is a costly but crucial step for understanding the kinetic relevance of distinct reaction pathways, product yields, and myriad other properties of reacting systems. The standard methodology for activation energy characterization has historically been a transition state search using the highest level of theory that can be afforded. However, recently, several groups have popularized the idea of predicting activation energies directly based on nothing more than the reactant and product graphs, a sufficiently complex neural network, and a broad enough data set. Here, we have revisited this task using the recently developed Reaction Graph Depth 1 (RGD1) transition state data set and several newly developed graph attention architectures. All of these new architectures achieve similar state-of-the-art results of ∼4 kcal/mol mean absolute error on withheld testing sets of reactions but poor performance on external testing sets composed of reactions with differing mechanisms, reaction molecularity, or reactant size distribution. Limited transferability is also shown to be shared by other contemporary graph to activation energy architectures through a series of case studies. We conclude that an array of standard graph architectures can already achieve results comparable to the irreducible error of available reaction data sets but that out-of-distribution performance remains poor.

Harary and hyper-Wiener indices of some graph operations

In this paper, we obtain the Harary index and the hyper-Wiener index of the H-generalized join of graphs and the generalized corona product of graphs. As a consequence, we deduce some of the results in (Das et al. in J. Inequal. Appl. 2013:339, 2013) and (Khalifeh et al. in Comput. Math. Appl. 56:1402–1407, 2008). Moreover, we calculate the Harary index and the hyper-Wiener index of the ideal-based zero-divisor graph of a ring.

Shallow Minors, Graph Products, and Beyond-Planar Graphs

Shallow Minors, Graph Products, and Beyond-Planar Graphs

Compactified Jacobians of extended ADE curves and Lagrangian fibrations

We observe that general reducible curves in sufficiently positive linear systems on K3 surfaces are of a form that generalize Kodaira’s classification of singular elliptic fibers and thus call them extended ADE curves. On such a curve [Formula: see text], we describe a compactified Jacobian and show that its components reflect the intersection graph of [Formula: see text]. This extends known results when [Formula: see text] is reduced, but new difficulties arise when [Formula: see text] is non-reduced. As an application, we get an explicit description of general singular fibers of certain Lagrangian fibrations of Beauville–Mukai type.