Special Graphs Research Articles

Labeled graph partitioning scheme for distributed edge caching

Distributed edge caching could address latency and congestion problems in large-scale data access effectively, improving system throughput and performance. However, the lack of specialized edge caching solutions for graph data resulted in low cache hit rates and high data access latency. This limitation stemmed from existing graph partitioning schemes that overlooked the specific requirements of edge caching, such as communication overhead between edge servers, data query frequency, and query patterns. To overcome these challenges, we proposed LGPE, a labeled graph partitioning scheme for distributed edge caching based on frequent query patterns. LGPE generated frequent query patterns according to the user’s historical query subgraphs and then performed labeled graph partitioning based on the frequent pattern, ensuring that the labeled graph divided into edge servers had a high hit rate for frequent user queries while satisfying a minimum edge cut. Evaluation on real datasets from various application domains demonstrated that LGPE achieved approximately 40% higher cache hit rates compared to benchmarks like METIS and SCOTCH, and it also performed well in terms of edge cuts in special graph cases.

Anti-Ramsey number of matchings in outerplanar graphs

A subgraph of an edge-colored graph is rainbow if all of its edges have different colors. Given a graph family G and a graph H, the anti-Ramsey number AR(G,H) is the maximum number of colors in an edge-coloring of a graph G∈G which has no rainbow copy of H. When G={G}, we write AR(G,H) for short. Erdős, Simonovits and Sós introduced these numbers in the 1970s, they studied the case when G is a complete graph. Since then, many results have appeared considering G and H belonging to specific graph classes. In particular, Jendrol’, Schiermeyer and Tu (Jendrol’ et al., 2014) obtained an upper bound for AR(Tn,kK2) in terms of n and k for k≥3 and n≥2k, where Tn is the family of maximal planar graphs on n vertices. Jin and Ye (Jin and Ye, 2018) continued the study of anti-Ramsey numbers of matchings in planar graphs, by deriving an upper bound for AR(Mn,kK2), where Mn is the family of maximal outerplanar graphs on n vertices. In both works, examples are shown for which there are colorings using many colors while avoiding rainbow matchings, but the numbers of colors in these examples do not reach the upper bounds. The bounds of AR(Tn,kK2) were improved in following works by other researchers. This paper goes in the same direction of these two works, proving exact value for AR(Bn,kK2) in terms of n and k for n≥6 and k≥3, where Bn is the family of maximal bipartite outerplanar graphs on n vertices which contains a Hamilton cycle.

A Graphical Representation of the Truncated Moment of the Solution of a Nonlinear SPDE

We consider a stochastic partial differential equation driven by a Lévy type noise (SPDE). Particular attention is given to the correlation function which measures the moments of the solution. Using the Feynman graph formalism, the solution of the SPDE as well as its truncated moments are given as a sum over specific graphs that are evaluated according to some rules. A remark on some applications will be given at the end of this work.

FNIRS-based graph frequency analysis to identify mild cognitive impairment in Parkinson’s disease

Background:Early identification of mild cognitive impairment (MCI) is essential for its treatment and the prevention of dementia in Parkinson’s disease (PD). Existing approaches are mostly based on neuropsychological assessments, while brain activation and connection have not been well considered. New Method:This paper presents a neuroimaging-based graph frequency analysis method and the generated features to quantify the brain functional neurodegeneration and distinguish between PD-MCI patients and healthy controls. The Stroop color-word experiment was conducted with 20 PD-MCI patients and 34 healthy controls, and the brain activation was recorded with functional near-infrared spectroscopy (fNIRS). Then, the functional brain network was constructed based on Pearson’s correlation coefficient calculation between every two fNIRS channels. Next, the functional brain network was represented as a graph and decomposed in the graph frequency domain through the graph Fourier transform (GFT) to obtain the eigenvector matrix. Total variation and weighted zero crossings of eigenvectors were defined and integrated to quantify functional interaction between brain regions and the spatial variability of the brain network in specific graph frequency ranges, respectively. After that, the features were employed in training a support vector machine (SVM) classifier. Results:The presented method achieved a classification accuracy of 0.833 and an F1 score of 0.877, significantly outperforming existing methods and features. Comparison with Existing Methods:Our method provided improved classification performance in the identification of PD-MCI. Conclusion:The results suggest that the presented graph frequency analysis method well identify PD-MCI patients and the generated features promise functional brain biomarkers for PD-MCI diagnosis.

Minimal toughness in special graph classes

Let $t$ be a positive real number. A graph is called $t$-tough if the removal of any vertex set $S$ that disconnects the graph leaves at most $|S|/t$ components, and all graphs are considered 0-tough. The toughness of a graph is the largest $t$ for which the graph is $t$-tough, whereby the toughness of complete graphs is defined as infinity. A graph is minimally $t$-tough if the toughness of the graph is $t$, and the deletion of any edge from the graph decreases the toughness. In this paper, we investigate the minimum degree and the recognizability of minimally $t$-tough graphs in the classes of chordal graphs, split graphs, claw-free graphs, and $2K_2$-free graphs.

Treewidth versus clique number. II. Tree-independence number

In 2020, we initiated a systematic study of graph classes in which the treewidth can only be large due to the presence of a large clique, which we call (tw,ω)-bounded. The family of (tw,ω)-bounded graph classes provides a unifying framework for a variety of very different families of graph classes, including graph classes of bounded treewidth, graph classes of bounded independence number, intersection graphs of connected subgraphs of graphs with bounded treewidth, and graphs in which all minimal separators are of bounded size. While Chaplick and Zeman showed in 2017 that (tw,ω)-bounded graph classes enjoy some good algorithmic properties related to clique and coloring problems, it is an interesting open problem to which extent (tw,ω)-boundedness has useful algorithmic implications for problems related to independent sets. We provide a partial answer to this question by identifying a sufficient condition for (tw,ω)-bounded graph classes to admit a polynomial-time algorithm for the Maximum Weight Independent Packing problem and, as a consequence, for the weighted variants of the Independent Set and Induced Matching problems.Our approach is based on a new min-max graph parameter related to tree decompositions. We define the independence number of a tree decomposition T of a graph as the maximum independence number over all subgraphs of G induced by some bag of T. The tree-independence number of a graph G is then defined as the minimum independence number over all tree decompositions of G. Boundedness of the tree-independence number is a refinement of (tw,ω)-boundedness that is still general enough to hold for all the aforementioned families of graph classes. Generalizing a result on chordal graphs due to Cameron and Hell from 2006, we show that if a graph is given together with a tree decomposition with bounded independence number, then the Maximum Weight Independent Packing problem can be solved in polynomial time. Applications of our general algorithmic result to specific graph classes are given in the third paper of the series [Dallard, Milanič, and Štorgel, Treewidth versus clique number. III. Tree-independence number of graphs with a forbidden structure].

Stability, Vertex Stability, and Unfrozenness for Special Graph Classes

Frei et al. (J. Comput. Syst. Sci. 123, 103–121, 2022) show that the stability, vertex stability, and unfrozenness problems with respect to certain graph parameters are complete for Θ2P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\\Theta _{2}^{\ extrm{P}}}$$\\end{document}, the class of problems solvable in polynomial time by parallel access to an NP oracle. They studied the common graph parameters α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\\alpha }$$\\end{document} (the independence number), β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\\beta }$$\\end{document} (the vertex cover number), ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\\omega }$$\\end{document} (the clique number), and χ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\\chi }$$\\end{document} (the chromatic number). We complement their approach by providing polynomial-time algorithms solving these problems for special graph classes, namely for graphs with bounded tree-width or bounded clique-width. In order to improve these general time bounds even further, we then focus on trees, forests, bipartite graphs, and co-graphs.

Innovative Macula Capillaries Plexuses Visualization with OCTA B-Scan Graph Representation: Transforming OCTA B-Scan into OCTA Graph Representation.

The aim of this study is to transform optical coherence tomography angiography (OCTA) scans into innovative OCTA graphs, serving as novel biomarkers representing the macular vasculature. The study included 90 healthy subjects and 39 subjects with various abnormalities (29 with diabetic retinopathy, 5 with age-related macular degeneration, and 5 with choroid neovascularization). OCTA 5µm macular coronal views (MCVs) were generated for each subject, followed by blood vessel segmentation and skeleton processing. Subsequently, the blood vessel density index, blood vessel skeleton index, and blood vessel tortuosity index were computed. The graphs of each metric were plotted against the axial axes of the OCTA B-scan, representing the integrity of vasculature at successive 5µm macular depths. The results revealed two significant findings. First, the B-scans from OCTA can be transformed into OCTA graphs, yielding three specific OCTA graphs in this study. These graphs provide new biomarkers for assessing the integrity of deep vascular complex (DVC) and superficial vascular complex (SVC) within the macula. Second, a statistically significant difference was observed between normal (n=90) and abnormal (n=39) subjects, with a t-test p-value significantly lower than 0.001. The Mann-Whitney u-test also yielded significant difference but only between the 90 normal and 29 DR subjects. The novel OCTA graphs offer a unique representation of the macula's SVC and DVC, suggesting their potential in aiding physicians in the diagnosis of eye health within OCTA clinics. Further research is warranted to finalize the shape of these newly derived OCTA graphs and establish their clinical relevance and utility.

Positive Consensus of Fractional-Order Multiagent Systems Over Directed Graphs.

This article investigates the positive consensus problem of a special kind of interconnected positive systems over directed graphs. They are composed of multiple fractional-order continuous-time positive linear systems. Unlike most existing works in the literature, we study this problem for the first time, in which the communication topology of agents is described by a directed graph containing a spanning tree. This is a more general and new scenario due to the interplay between the eigenvalues of the Laplacian matrix and the controller gains, which renders the positivity analysis fairly challenging. Based on the existing results in spectral graph theory, fractional-order systems (FOSs) theory, and positive systems theory, we derive several necessary and/or sufficient conditions on the positive consensus of fractional-order multiagent systems (PCFMAS). It is shown that the protocol, which is designed for a specific graph, can solve the positive consensus problem of agents over an additional set of directed graphs. Finally, a comprehensive comparison study of different approaches is carried out, which shows that the proposed approaches have advantages over the existing ones.

Demystifying Graph Sparsification Algorithms in Graph Properties Preservation

Graph sparsification is a technique that approximates a given graph by a sparse graph with a subset of vertices and/or edges. The goal of an effective sparsification algorithm is to maintain specific graph properties relevant to the downstream task while minimizing the graph's size. Graph algorithms often suffer from long execution time due to the irregularity and the large real-world graph size. Graph sparsification can be applied to greatly reduce the run time of graph algorithms by substituting the full graph with a much smaller sparsified graph, without significantly degrading the output quality. However, the interaction between numerous sparsifiers and graph properties is not widely explored, and the potential of graph sparsification is not fully understood. In this work, we cover 16 widely-used graph metrics, 12 representative graph sparsification algorithms, and 14 real-world input graphs spanning various categories, exhibiting diverse characteristics, sizes, and densities. We developed a framework to extensively assess the performance of these sparsification algorithms against graph metrics, and provide insights to the results. Our study shows that there is no one sparsifier that performs the best in preserving all graph properties, e.g. sparsifiers that preserve distance-related graph properties (eccentricity) struggle to perform well on Graph Neural Networks (GNN). This paper presents a comprehensive experimental study evaluating the performance of sparsification algorithms in preserving essential graph metrics. The insights inform future research in incorporating matching graph sparsification to graph algorithms to maximize benefits while minimizing quality degradation. Furthermore, we provide a framework to facilitate the future evaluation of evolving sparsification algorithms, graph metrics, and ever-growing graph data.

On the Local Metric Dimension of Line Graphs

Let [Formula: see text] be a graph. For any [Formula: see text], if there exists [Formula: see text] such that [Formula: see text], we say that [Formula: see text] resolving [Formula: see text]. A set [Formula: see text] of vertices in [Formula: see text] is a local resolving set of [Formula: see text] if there exists [Formula: see text] such that [Formula: see text] for any [Formula: see text]. The local metric dimension [Formula: see text] of [Formula: see text] is the minimum cardinality of all the local resolving sets of [Formula: see text]. In this paper, we study the relation between [Formula: see text] and [Formula: see text]. Furthermore, we construct a graph [Formula: see text] such that [Formula: see text] and [Formula: see text]. Finally, we investigate the local metric dimension of several special line graphs.

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.

Outer-Convex Hop Domination in Graphs Under Some Binary Operations

Let G be a graph with vertex and edge sets V (G) and E(G), respectively. A set C ⊆ V (G) is called an outer-convex hop dominating if for every two vertices x, y ∈ V (G) \ C, the vertex set of every x−y geodesic is contained in V (G) \ C and for every a ∈ V (G) \ C, there exists b ∈ C such that dG(a, b) = 2. The minimum cardinality of an outer-convex hop dominating set of G, denoted by ̃γconh(G), is called the outer-convex hop domination number of G. In this paper, we generate some formulas for the parameters of some special graphs and graphs under some binary operations by characterizing first the outer-convex hop dominating sets of each of thesegraphs. Moreover, we establish realization result that identifies and determines the connection of this parameter with the standard hop domination parameter. It shows that given any graph, this new parameter is always greater than or equal to the standard hop domination parameter.

$J^2$-Hop Domination in Graphs: Properties and Connections with Other Parameters

A subset $T=\{v_1, v_2, \cdots, v_m\}$ of vertices of an undirected graph $G$ is called a $J^2$-set if$N_G^2[v_i]\setminus N_G^2[v_j]\neq \varnothing$ for every $i\neq j$, where $i,j\in\{1, 2, \ldots, m\}$.A $J^2$-set is called a $J^2$-hop dominating in $G$ if for every $a\in V(G)\s T$, there exists $b\in T$ such that $d_G(a,b)=2$. The $J^2$-hop domination number of $G$, denoted by $\gamma_{J^2h}(G)$, is the maximum cardinality among all $J^2$-hop dominating sets in $G$. In this paper, we introduce this new parameter and wedetermine its connections with other known parameters in graph theory. We derive its bounds with respect to the order of a graph and other known parameters on a generalized graph, join and corona of two graphs. Moreover,we obtain exact values of the parameter for some special graphs and shadow graphs using the characterization results that are formulated in this study.

Certified Perfect Domination in Graphs

Let $ G = (V(G),E(G)) $ be a simple connected graph. A set $ S \subseteq V(G) $ is called a certified perfect dominating set of $ G $ if every vertex $ v \in V(G)\setminus S $ is dominated by exactly one element $ u \in S $, such that $ u $ has either zero or at least two neighbors in $ V(G)\setminus S $. The minimum cardinality of a certified perfect dominating set of $ G $ is called the \textit{certified perfect domination number} of $ G $ and denoted by $ \gamma_{cerp}(G) $. A certified perfect dominating set $ S $ of $ G $ with $ \lvert S \rvert = \gamma_{cerp}(G) $ is called a $ \gamma_{cerp} $-set. In this paper, the author focuses on several key aspects: a characterization of the certified perfect dominating set, determining the exact values of the certified perfect domination number for specific graphs, and investigating the certified perfect domination number of graphs resulting from the join of two graphs. Furthermore, some relationships between the certified dominating set, the perfect dominating set, and the certified perfect dominating set of a graph $ G $ are established.

The energy complexity of diameter and minimum cut computation in bounded-genus networks

This paper investigates the energy complexity of distributed graph problems in multi-hop radio networks, where the energy cost of an algorithm is measured by the maximum number of awake rounds of a vertex. Recent works revealed that some problems, such as broadcast, breadth-first search, and maximal matching, can be solved with energy-efficient algorithms that consume only polylog⁡n energy. However, there exist some problems, such as computing the diameter of the graph, that require Ω(n) energy to solve. To improve energy efficiency for these problems, we focus on a special graph class: bounded-genus graphs. We present algorithms for computing the exact diameter, the exact global minimum cut size, and a (1±ϵ)-approximate s-t minimum cut size with O˜(n) energy for bounded-genus graphs. Our approach is based on a generic framework that divides the vertex set into high-degree and low-degree parts and leverages the structural properties of bounded-genus graphs to control the number of certain connected components in the subgraph induced by the low-degree part.

Result on The Weak Non-Split Independent Domination Number of Some Special Graphs

A graph has a vertex set and edge set and . A non - empty subset D is an independent dominating set if every vertex in is adjacent a vertex in and is a non – adjacent vertices. The domination number is the minimum cardinality of an independent dominating set. If, in addition let and be the element of . Then weakly dominates if (i) and (ii) . A set S is a weak non-split dominating set of if every vertex is is weakly dominated by at least one vertex in and the induced subgraph is connected. The minimum cardinality of a weak non-split independent dominating set is the weak non-split independent domination number of G .The main purpose of this paper is to introduce the concept of weak non-split independent domination number. For that we have chosen Soifer graph, Chvatal graph, Fritsch graph, Herschel graph, Moser graph, Franklin graph to find the weak non-split domination number.

Regular divisor graph of finite commutative ring

Let R be a finite commutative ring with identity 1. We introduce a new graph called regular divisor graph and denoted by . We classify the finite commutative ring to get a special graph and we are going to study some properties of this graph, clique number, chromatic number, number of cycles, connectivity and blocks.

Graceful labeling construction for some special tree graph using adjacency matrix

Graceful labeling construction for some special tree graph using adjacency matrix

Research on the Method of Hypergraph Construction of Information Systems Based on Set Pair Distance Measurement

As a kind of special graph of structured data, a hypergraph can intuitively describe not only the higher-order relation and complex connection mode between nodes but also the implicit relation between nodes. Aiming at the limitation of traditional distance measurement in high-dimensional data, a new method of hypergraph construction based on set pair theory is proposed in this paper. By means of dividing the relationship between data attributes, the set pair connection degree between samples is calculated, and the set pair distance between samples is obtained. Then, on the basis of set pair distance, the combination technique of k-nearest neighbor and ε radius is used to construct a hypergraph, and high-dimensional expression and hypergraph clustering are demonstrated experimentally. By performing experiments on different datasets on the Kaggle open-source dataset platform, the comparison of cluster purity, the Rand coefficient, and normalized mutual information are shown to demonstrate that this distance measurement method is more effective in high-dimensional expression and exhibits a more significant performance improvement in spectral clustering.