Search Results (57)

This paper introduces a framework of hypercompositional algebra on fuzzy graphs by defining and analyzing fuzzy path-based hyperoperations. Building on the notion of strongest strong paths (paths that are both strength-optimal and composed exclusively of strong edges, where each edge achieves maximum connection strength between its endpoints), we define two operations: a vertex-based fuzzy path hyperoperation and an edge-based variant. These operations generalize classical graph hyperoperations to the fuzzy setting while maintaining compatibility with the underlying topology. We prove that the vertex fuzzy path hyperoperation is associative, forming a fuzzy hypersemigroup, and establish additional properties such as reflexivity and monotonicity with respect to $\alpha$-cuts. Structural features such as fuzzy strong cut vertices and edges are examined, and a fuzzy distance function is introduced to quantify directional connectivity strength. We define an equivalence relation based on mutual full-strength reachability and construct a quotient fuzzy graph that reflects maximal closed substructures under the vertex fuzzy path hyperoperation. Applications are discussed in domains such as trust networks, biological systems, and uncertainty-aware communications. This work aims to lay the algebraic foundations for further exploration of fuzzy hyperstructures that support modeling, analysis, and decision-making in systems governed by partial and asymmetric relationships. Full article

In recent years, wireless sensor networks (WSNs) have been employed across various domains, including military services, healthcare, disaster response, industrial automation, and smart infrastructure. Due to the absence of fixed communication infrastructure, WSNs rely on ad hoc connections between sensor nodes to transmit sensed data to target nodes. Within a WSN, a sensor node whose failure partitions the network into disconnected segments is referred to as a critical node or cut vertex. Identifying such nodes is a fundamental step toward ensuring the reliability of WSNs. The critical node detection problem (CNDP) focuses on determining the set of nodes whose removal most significantly affects the network’s connectivity, stability, functionality, robustness, and resilience. CNDP is a significant challenge in network analysis that involves identifying the nodes that have a significant influence on connectivity or centrality measures within a network. However, achieving an optimal solution for the CNDP is often hindered by its time-consuming and computationally intensive nature, especially when dealing with large-scale networks. In response to this challenge, we present a method based on particle swarm optimization (PSO) for the detection of critical nodes. We employ discrete PSO (DPSO) along with the modified position equation (MPE) to effectively solve the CNDP, making it applicable to various k-vertex variations of the problem. We examine the impact of population size on both execution time and result quality. Experimental analysisusing different neighborhood topologies—namely, the star topology and the dynamic topology—was conducted to analyze their impact on solution effectiveness and adaptability to diverse network configurations. We consistently observed better result quality with the dynamic topology compared to the star topology for the same population size, while the star topology exhibited better execution time. Our findings reveal the promising efficacy of the proposed solution in addressing the CNDP, achieving high-quality solutions compared to existing methods. Full article

The Maximum Colored Cut problem aims to seek a bipartition of the vertex set of a graph, maximizing the number of colors in the crossing edges. It is a classical Max-Cut problem if the host graph is rainbow. Let $mcc\left(G\right)$ denote the maximum number of colors in a cut of an edge-colored graph G. Let $C_k$ be a cycle of length k; we say G is PC-$C_k$-free if G contains no properly colored $C_k$. We say G is a p-edge-colored graph if there exist p colors in G. In this paper, we first show that if G is a PC-$C_3$-free p-edge-colored complete 4-partite graph, then $mcc\left(G\right)=p$. Let $k\ge 3$ be an integer. Then, we show that if G is a PC-$C_4$-free p-edge-colored complete k-partite graph, then $mcc\left(G\right)\ge min\{p-1,15p/16\}$. Finally, for a p-edge-colored complete graph G, we prove that $mcc\left(G\right)\ge p-1$ if G is PC-$C_4$-free, and $mcc\left(G\right)\ge min\{p-6,7p/8\}$ if G is PC-$C_5$-free and $p\ge 7$. Full article

In this paper, a circuit-partitioning method is proposed based on partition connectivity clustering and tabu search. It includes four phases: coarsening, initial partitioning, uncoarsening, and refinement. In the initial partitioning phase, the concept of partition connectivity is introduced to optimize the vertex-clustering process, which clusters vertices with high connectivity in advance to provide an optimal initial solution. In the refinement phase, an improved tabu search algorithm is proposed, which combines two flexible neighborhood search rules and a candidate solution-selection strategy based on vertex-exchange frequency to further optimize load balancing. Additionally, a random perturbation method is suggested to increase the diversity of the search space and improve both the depth and breadth of global search. The experimental results based on the ISCAS-89 and ISCAS-85 benchmark circuits show that the average cut size of the proposed circuit-partitioning method is 0.91 times that of METIS and 0.86 times that of the KL algorithm, with better performance for medium- and small-scale circuits. Full article

In this paper, we propose an unstructured cut-cell generation method for complex geological modeling. The method can robustly and quickly generate cut results for surface and polyhedral meshes. First, we correctly identify intersecting elements in the input and compute intersection points and lines. Then, we integrate the intersection points and lines into the mesh face and subdivide it into a set of triangles. Finally, each mesh element is considered to be inside or outside each input object, and the result is finally extracted from the mesh elements generated in the above steps. To support topological queries and modifications in cutting process, we design a novel polyhedral mesh data structure, which introduces the concept of half-edge but represents it in an implicit manner. For each cell, we record its incident faces. For each face, we store the incident half-edges. For each vertex and edge, we store one of its incident faces. Our method is properly proved in a complex 3D geological model. Full article

Vertex models have become essential tools for understanding tissue morphogenesis by simulating the mechanical and geometric properties of cells in various biological systems. These models represent cells as polygons or polyhedra, capturing cellular interactions such as adhesion, tension, and force generation. This review explores the ongoing evolution of computational vertex models, highlighting their application to complex tissue dynamics, including organoid development, wound healing, and cancer metastasis. We examine different energy formulations used in vertex models, which account for mechanical forces such as surface tension, volume conservation, and intercellular adhesion. Additionally, this review discusses the challenges of expanding traditional 2D models to 3D structures, which require the inclusion of factors like mechanical polarisation and topological transitions. We also introduce recent advancements in modelling techniques that allow for more flexible and dynamic cell shapes, addressing limitations in earlier frameworks. Mechanochemical feedback and its role in tissue behaviour are explored, along with cutting-edge approaches like self-propelled Voronoi models. Finally, the review highlights the importance of parameter inference in these models, particularly through Bayesian methods, to improve accuracy and predictive power. By integrating these new insights, vertex models continue to provide powerful frameworks for exploring the complexities of tissue morphogenesis. Full article

Closeness is a measure that quantifies how quickly information can spread from a given node to all other nodes in the network, reflecting the efficiency of communication within the network by indicating how close a node is to all other nodes. For a graph G, the subset S of vertices of $V\left(G\right)$ is called vertex cut of G if the graph $G-S$ becomes disconnected. The minimum cardinality of S for which $G-S$ is either disconnected or contains precisely one vertex is called connectivity of G. A graph is called k-connected if it stays connected even when any set of fewer than k vertices is removed. In communication networks, a k-connected graph improves network reliability; even if up to $k-1$ nodes fail, the network remains operational, maintaining connectivity between devices. This paper aims to study the concept of closeness within n-vertex graphs with fixed connectivity. First, we identify the graphs that maximize the closeness among all graphs of order n with fixed connectivity k. Then, we determine the graphs that achieve the maximum closeness within all k-connected graphs of order n, given specific fixed parameters such as diameter, independence number, and minimum degree. Full article

One nappe of a right circular cone, cut by a transverse plane, splits the cone into an infinite frustum and a cone with an elliptical section of finite volume. There is a standard way of computing this finite volume, which involves finding the parameters of the so-called shadow ellipse, the characteristics of the oblique ellipse (the cut) and, finally, the projection of the vertex of the cone onto the oblique ellipse. This paper shows that it is possible to compute that volume just by using the information of the shadow ellipse and the height of the cone. Indeed, the finite slant cone has the same volume of an elliptic right cone, with the base being the shadow ellipse of the cut portion and with the height being the distance between the vertex of the cone and the intersection of the height of the original cone with the cutting plane. This is proved by introducing a volume-preserving shear transformation of the elliptical slant cone to a right cone, so that the standard volume formula for a cone can be straightforwardly applied. This implies a simplification in the procedure for computing the volume, since the oblique ellipse—i.e., the difficult part—can be neglected because only the shadow ellipse needs to be determined. Full article

The restricted edge-connectivity of a connected graph G, denoted by ${\lambda }^{\prime }\left(G\right)$, 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 ${\lambda }^{\prime }\left(G\right)$ 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 ${\lambda }^{\prime }\left(G\right)=\xi \left(G\right)$, 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 $V\left(G\right)\times V\left(H\right)$ and the edge set $\{(x_1,y_1)(x_2,y_2)|x_1=x_2$ and $y_1{y}_2\in E\left(H\right)$; or $y_1=y_2$ and $x_1{x}_2\in E\left(G\right)$; or $x_1{x}_2\in E\left(G\right)$ and $y_1{y}_2\in E\left(H\right)$}. In this paper, we determine, for any nontrivial connected graph G, the restricted edge-connectivity of $G⊠P_n$, $G⊠C_n$ and $G⊠K_n$, where $P_n$, $C_n$ and $K_n$ are the path, cycle and complete graph of order n, respectively. As corollaries, we give sufficient conditions for these strong product graphs $G⊠P_n$, $G⊠C_n$ and $G⊠K_n$ to be maximally restricted edge-connected and super restricted edge-connected. Full article

Vague influence graphs (VIGs) are well articulated, useful and practical tools for managing the uncertainty preoccupied in all real-life difficulties where ambiguous facts, figures and explorations are explained. A VIG gives the information about the effect of a vertex on the edge. In this paper, we present the domination concept for VIG. Some issues and results of the domination in vague graphs (VGs) are also developed in VIGs. We defined some basic notions in the VIGs such as the walk, path, strength of $\mathcal{I}n$-pair , strong $\mathcal{I}n$-pair, $\mathcal{I}n$-cut vertex, $\mathcal{I}n$-cut pair (CP), complete VIG and strong pair domination number in VIG. Finally, an application of domination in illegal drug trade was introduced. Full article

Background: As advanced technology continues to evolve, incorporating robotics into surgical procedures has become imperative for precision and accuracy in preoperative planning. Nevertheless, the integration of three-dimensional (3D) imaging into these processes presents both financial considerations and potential patient safety concerns. This study aims to assess the accuracy of a novel 2D-to-3D knee reconstruction solution, RSIP XPlan.ai™ (RSIP Vision, Jerusalem, Israel), on preoperative total knee arthroplasty (TKA) patient anatomies. Methods: Accuracy was calculated by measuring the Root Mean Square Error (RMSE) between X-ray-based 3D bone models generated by the algorithm and corresponding CT bone segmentations (distances of each mesh vertex to the closest vertex in the second mesh). The RMSE was computed globally for each bone, locally for eight clinically relevant bony landmark regions, and along simulated bone cut contours. In addition, the accuracies of three anatomical axes were assessed by comparing angular deviations to inter- and intra-observer baseline values. Results: The global RMSE was 0.93 ± 0.25 mm for the femur and 0.88 ± 0.14 mm for the tibia. Local RMSE values for bony landmark regions were 0.51 ± 0.33 mm for the five femoral landmarks and 0.47 ± 0.17 mm for the three tibial landmarks. The RMSE along simulated cut contours was 0.75 ± 0.35 mm for the distal femur cut and 0.63 ± 0.27 mm for the proximal tibial cut. Anatomical axial average angular deviations were 1.89° for the trans epicondylar axis (with an inter- and intra-observer baseline of 1.43°), 1.78° for the posterior condylar axis (with a baseline of 1.71°), and 2.82° (with a baseline of 2.56°) for the medial–lateral transverse axis. Conclusions: The study findings demonstrate promising results regarding the accuracy of XPlan.ai™ in reconstructing 3D bone models from plain-film X-rays. The observed accuracy on real-world TKA patient anatomies in anatomically relevant regions, including bony landmarks, cut contours, and axes, suggests the potential utility of this method in various clinical scenarios. Further validation studies on larger cohorts are warranted to fully assess the reliability and generalizability of our results. Nonetheless, our findings lay the groundwork for potential advancements in future robotic arthroplasty technologies, with XPlan.ai™ offering a promising alternative to conventional CT scans in certain clinical contexts. Full article

We explore strong Nash equilibria in the max k-cut game on an undirected and unweighted graph with a set of k colors. Here, the vertices represent players, and the edges denote their relationships. Each player, v, selects a color as its strategy, and its payoff (or utility) is determined by the number of neighbors of v who have chosen a different color. Limited findings exist on the existence of strong equilibria in max k-cut games. In this paper, we make advancements in understanding the characteristics of strong equilibria. Specifically, our primary result demonstrates that optimal solutions are seven-robust equilibria. This implies that for a coalition of vertices to deviate and shift the system to a different configuration, i.e., a different coloring, a number of coalition vertices greater than seven is necessary. Then, we establish some properties of the minimal subsets concerning a robust deviation, revealing that each vertex within these subsets will deviate toward the color of one of its neighbors. Full article

A vertex degree based topological index called the Sombor index was recently defined in 2021 by Gutman and has been very popular amongst chemists and mathematicians. We determine the amount of change of the Sombor index when some elements are removed from a graph. This is done for several graph elements, including a vertex, an edge, a cut vertex, a pendant edge, a pendant path, and a bridge in a simple graph. Also, pendant and non-pendant cases are studied. Using the obtained formulae successively, one can find the Sombor index of a large graph by means of the Sombor indices of smaller graphs that are just graphs obtained after removal of some vertices or edges. Sometimes, using iteration, one can manage to obtain a property of a really large graph in terms of the same property of many other subgraphs. Here, the calculations are made for a pendant and non-pendant vertex, a pendant and non-pendant edge, a pendant path, a bridge, a bridge path from a simple graph, and, finally, for a loop and a multiple edge from a non-simple graph. Using these results, the Sombor index of cyclic graphs and tadpole graphs are obtained. Finally, some Nordhaus–Gaddum type results are obtained for the Sombor index. Full article

This paper introduces a new kind of quasicrystal by Fibonacci-spacing a multigrid of a certain symmetry, like $H_2$, $H_3$, $T_3$, etc. Multigrids of a certain symmetry can be used to generate quasicrystals, but multigrid vertices are not a quasicrystal due to arbitrary closeness. By Fibonacci-spacing the grids, the structure transit into an aperiodic order becomes a quasicrystal itself. Unlike the quasicrystal generated by the dual-grid method, this kind of quasicrystal does not live in the dual space of the grid space. It is the grid space itself and possesses quasicrystal properties, except that its total number of vertex types are not finite and fixed for the infinite size of the quasicrystal but bounded by a slowly logarithmic growing number. A 2D example, the Fibonacci pentagrid, is given. A 3D example, the Fibonacci icosagrid (FIG), is also introduced, as well as its subsets, the Fibonacci tetragrid (FTG). The FIG can be thought of as a golden composition of five sets of FTGs. The golden composition procedure is another way to transit a random structure into aperiodic order, and the associated rotational angle is the same as the angle that resolves the geometric frustration for the $H_3$ tetrahedral clusters. The FIG resembles another quasicrystal that is the same golden composition of five quasicrystals that are cut and projected and sliced from the $E_8$ lattice. This leads to further exploration in mapping the FIG to the $E_8$ lattice, and the results will be published following this paper. Full article

Connectivity plays an important role in measuring the fault tolerance of interconnection networks. As a special class of connectivity, m-component connectivity is a natural generalization of the traditional connectivity of graphs defined in terms of the minimum vertex cut. Moreover, it is a more advanced metric to assess the fault tolerance of a graph G. Let $G=\left(V\right(G),E(G\left)\right)$ be a non-complete graph. A subset F$(F\subseteq V(G\left)\right)$ is called an m-component cut of G, if $G-F$ is disconnected and has at least m components $(m\ge 2)$. The m-component connectivity of G, denoted by $c{\kappa }_m\left(G\right)$, is the cardinality of the minimum m-component cut. Let $C{F}_n$ denote the n-dimensional leaf-sort graph. Since many structures do not exist in leaf-sort graphs, many of their properties have not been studied. In this paper, we show that $c{\kappa }_3\left(C{F}_n\right)=3n-6$ $(n$ is odd) and $c{\kappa }_3\left(C{F}_n\right)=3n-7$ $(n$ is even) for $n\ge 3$; $c{\kappa }_4\left(C{F}_n\right)=\frac{9n-21}{2}$ $(n$ is odd) and $c{\kappa }_4\left(C{F}_n\right)=\frac{9n-24}{2}$ $(n$ is even) for $n\ge 4$. Full article