Search Results (40)

A graph is called strongly Menger-edge connected (SME-connected) if any two vertices are connected by as many edge-disjoint paths as their smaller degree. For positive integers t and r, a graph G is called t-edge-fault-tolerant SME-connected (t-EFT-SME-connected) of order r if $G-F$ is SME-connected for any set F of edges in G with $\left|F\right|\le t$ and $\delta (G-F)\ge r$. We show that the n-dimensional folded crossed cube is $(n-1)$-EFT-SME-connected of order 1 and $(3n-5)$-EFT-SME-connected of order 2. Let $p(G,f)$ and $p^M(G,f)$ be the probabilities that G is connected and SME-connected when f edges are faulted randomly, respectively. We perform a numerical simulation on $p(G,f)$ and $p^M(G,f)$ for a five-dimensional folded crossed cube and folded hypercube. The numerical results show that, in addition to their same edge connectivity and SME connectivity, these two graphs have almost the same values of $p(G,f)$ and $p^M(G,f)$ for every f. This hints that, although the ‘edge-cross’ pattern in a hypercube-based graph can shorten the mean vertex distance, the ‘edge-cross’ is not a necessary pattern for strengthening the connectivity of the graph. Full article

Tree-width and path-width are well-known graph parameters. Many NP-hard graph problems admit polynomial-time solutions when restricted to graphs of bounded tree-width or bounded path-width. In this work, we study the behavior of tree-width and path-width under various unary and binary graph transformations. For considered transformations, we provide upper and lower bounds for the tree-width and path-width of the resulting graph in terms of those of the initial graphs or argue why such bounds are impossible to specify. Among the studied unary transformations are vertex addition, vertex deletion, edge addition, edge deletion, subgraphs, vertex identification, edge contraction, edge subdivision, minors, powers of graphs, line graphs, edge complements, local complements, Seidel switching, and Seidel complementation. Among the studied binary transformations, we consider the disjoint union, join, union, substitution, graph product, 1-sum, and corona of two graphs. Full article

The Ramsey number $R\left(F\right)$ of a graph F without isolated vertices is the smallest positive integer n such that every red–blue coloring of $K_n$ produces a subgraph isomorphic to F all of whose edges are colored the same. Let $\mathcal{F}$ be a set of graphs without isolated vertices. For a positive integer t, the vertex-disjoint Ramsey number $V{R}_t\left(\mathcal{F}\right)$ is the smallest positive integer n such that every red–blue coloring of the complete graph $K_n$ of order n results in at least t pairwise vertex-disjoint monochromatic graphs in $\mathcal{F}$; while the edge-disjoint Ramsey number $E{R}_t\left(\mathcal{F}\right)$ is the smallest positive integer n such that every red–blue coloring of $K_n$ produces at least t pairwise edge-disjoint monochromatic graphs in $\mathcal{F}$. If $t=1$ and $\mathcal{F}$ consists of a single graph F, then $V{R}_1\left(\mathcal{F}\right)=E{R}_1\left(\mathcal{F}\right)=R\left(F\right)$ is the Ramsey number of the graph F. Thus, the concepts of vertex-disjoint and edge-disjoint Ramsey numbers provide a generalization of the standard Ramsey number. Upper and lower bounds for $V{R}_t\left(\mathcal{F}\right)$ and $E{R}_t\left(\mathcal{F}\right)$ are established for sets $\mathcal{F}$ of graphs without isolated vertices and the sharpness of these bounds is discussed. The primary goal of this paper is to investigate the values of $V{R}_t\left(\mathcal{F}\right)$ and $E{R}_t\left(\mathcal{F}\right)$ for sets $\mathcal{F}$ of graphs of size 2 or 3 without isolated vertices. The exact values of $V{R}_t\left(\mathcal{F}\right)$ are determined for all such sets $\mathcal{F}$ and all integers $t\ge 2$. The exact values of $E{R}_t\left(\mathcal{F}\right)$ of certain such sets $\mathcal{F}$ with prescribed conditions for all integers $t\ge 2$ are determined. For some special sets $\mathcal{F}$ of graphs of size 2 or 3 without isolated vertices, the exact values of $E{R}_t\left(\mathcal{F}\right)$ are determined for $2\le t\le 4$. Additional results, problems, and conjectures are also presented dealing with these two Ramsey concepts for graphs in general. Full article

Prompt and precise cropland mapping is indispensable for safeguarding food security, enhancing land resource utilization, and advancing sustainable agricultural practices. Conventional approaches faced difficulties in complex terrain marked by fragmented plots, pronounced elevation differences, and non-uniform field borders. To address these challenges, we propose DAENet, a novel deep learning framework designed for accurate cropland extraction from high-resolution GaoFen-1 (GF-1) satellite imagery. DAENet employs a novel Geometric-Optimized and Boundary-Restrained (GOBR) Block, which combines channel attention, multi-scale spatial attention, and boundary supervision mechanisms to effectively mitigate challenges arising from disjointed cropland parcels, topography-cast shadows, and indistinct edges. We conducted comparative experiments using 8 mainstream semantic segmentation models. The results demonstrate that DAENet achieves superior performance, with an Intersection over Union (IoU) of 0.9636, representing a 4% improvement over the best-performing baseline, and an F1-score of 0.9811, marking a 2% increase. Ablation analysis further validated the indispensable contribution of GOBR modules in improving segmentation precision. Using our approach, we successfully extracted 25,556.98 hectares of cropland within the study area, encompassing a total of 67,850 individual blocks. Additionally, the proposed method exhibits robust generalization across varying spatial resolutions, underscoring its effectiveness as a high-accuracy solution for agricultural monitoring and sustainable land management in complex terrain. Full article

It is well known that the famous Ramsey number $R(K_3,K_3)=6$. That is, the minimum positive integer n for which every red-blue coloring of the edges of the complete graph $K_n$ results in a monochromatic triangle $K_3$ is 6. It is also known that every red-blue coloring of $K_6$ results in at least two monochromatic triangles, which need not be vertex-disjoint or edge-disjoint. This fact led to an extension of Ramsey numbers. For a graph F and a positive integer t, the vertex-disjoint Ramsey number $V{R}_t\left(F\right)$ is the minimum positive integer n such that every red-blue coloring of the edges of the complete graph $K_n$ of order n results in t pairwise vertex-disjoint monochromatic copies of subgraphs isomorphic to F, while the edge-disjoint Ramsey number $E{R}_t\left(F\right)$ is the corresponding number for edge-disjoint subgraphs. Since $V{R}_1\left(F\right)$ and $E{R}_1\left(F\right)$ are the well-known Ramsey numbers of F, these new Ramsey concepts generalize the Ramsey numbers and provide a new perspective for this classical topic in graph theory. These numbers have been investigated for the two connected graphs $K_3$ and the path $P_3$ of order 3. Here, we study these numbers for the remaining connected graphs, namely, the path $P_4$ and the star $K_{1,3}$ of size 3. We show that $V{R}_t\left(P_4\right)=4t+1$ for every positive integer t and $V{R}_t\left(K_{1,3}\right)=4t$ for every integer $t\ge 2$. For $t\le 4$, the numbers $E{R}_t\left(K_{1,3}\right)$ and $E{R}_t\left(P_4\right)$ are determined. These numbers provide information towards the goal of determining how the numbers $V{R}_t\left(F\right)$ and $E{R}_t\left(F\right)$ increase as t increases for each graph $F\in \{K_{1,3},P_4\}$. Full article

A fundamental aspect of network analysis involves pinpointing nodes that hold significant positions within the network. Graph theory has emerged as a powerful mathematical tool for this purpose, and there exist numerous graph-theoretic parameters for analyzing the stability of the system. Within this framework, various graph-theoretic parameters contribute to network analysis. One such parameter used in network analysis is the so-called closeness, which serves as a structural measure to assess the efficiency of a node’s ability to interact with other nodes in the network. Mathematically, it measures the reciprocal of the sum of the shortest distances from a node to all other nodes in the network. A bipartite network is a particular type of network in which the nodes can be divided into two disjoint sets such that no two nodes within the same set are adjacent. This paper mainly studies the problem of determining the network that maximize the closeness within bipartite networks. To be more specific, we identify those networks that maximize the closeness over bipartite networks with a fixed number of nodes and one of the fixed parameters: connectivity, dissociation number, cut edges, and diameter. Full article

For $S\subseteq V\left(G\right),{\kappa }_G\left(S\right)$ denotes the maximum number k of edge disjoint trees $T_1,T_2,\dots ,T_k$ in G, such that $V\left(T_i\right)\cap V\left(T_j\right)=S$ for any $i,j\in \{1,2,\dots ,k\}$ and $i\ne j$. For an integer $2\le r\le \left|V\right(G\left)\right|$, the generalized r-connectivity of G is defined as ${\kappa }_r\left(G\right)=min\{{\kappa }_G\left(S\right)|S\subseteq V\left(G\right)\mathrm{and}|S|=r\}$. In fact, ${\kappa }_2\left(G\right)$ is the traditional connectivity of G. Hence, the generalized r-connectivity is an extension of traditional connectivity. The exchanged folded hypercube $EFH(s,t)$, in which $s\ge 1$ and $t\ge 1$ are positive integers, is a variant of the hypercube. In this paper, we find that ${\kappa }_3\left(EFH(s,t)\right)=s+1$ with $3\le s\le t$. Full article

The comparison of multiple (labeled) graphs with unrelated vertex sets is an important task in diverse areas of applications. Conceptually, it is often closely related to multiple sequence alignments since one aims to determine a correspondence, or more precisely, a multipartite matching between the vertex sets. There, the goal is to match vertices that are similar in terms of labels and local neighborhoods. Alignments of sequences and ordered forests, however, have a second aspect that does not seem to be considered for graph comparison, namely the idea that an alignment is a superobject from which the constituent input objects can be recovered faithfully as well-defined projections. Progressive alignment algorithms are based on the idea of computing multiple alignments as a pairwise alignment of the alignments of two disjoint subsets of the input objects. Our formal framework guarantees that alignments have compositional properties that make alignments of alignments well-defined. The various similarity-based graph matching constructions do not share this property and solve substantially different optimization problems. We demonstrate that optimal multiple graph alignments can be approximated well by means of progressive alignment schemes. The solution of the pairwise alignment problem is reduced formally to computing maximal common induced subgraphs. Similar to the ambiguities arising from consecutive indels, pairwise alignments of graph alignments require the consideration of ambiguous edges that may appear between alignment columns with complementary gap patterns. We report a simple reference implementation in Python/NetworkX intended to serve as starting point for further developments. The computational feasibility of our approach is demonstrated on test sets of small graphs that mimimc in particular applications to molecular graphs. Full article

Time-Sensitive Networking (TSN) and edge computing are promising networking technologies for the future of the Industrial Internet. TSN provides a reliable and deterministic low-latency communication service for edge computing. The Frame Replication and Elimination for Reliability (FRER) mechanism is important for improving the network reliability of TSN. It achieves high reliability by transmitting identical frames in parallel on two disjoint paths, while eliminating duplicated frames at the destination node. However, there are two problems with the FRER mechanism. One problem is that it does not consider the path reliability, and the other one is that it is difficult to find two completely disjoint path pairs in some cases. To solve the above problems, this paper proposes a method to find edge-disjoint path pairs considering path reliability for FRER in TSN. The method includes two parts: one is building a reliability model for paths, and the other one is computing a working path and a redundant path with the Edge-Disjoint Path Pairs Selection (EDPPS) algorithm. Theoretical and simulation results show that the proposed method effectively improves path reliability while reducing the delay jitter of frames. Compared with the traditional FRER mechanism, the proposed method reduces delay jitter by 15.6% when the network load is 0.9. Full article

A $(2,6)$-fullerene F is a 2-connected cubic planar graph whose faces are only $2$-length and $6$-length. Furthermore, it consists of exactly three $2$-length faces by Euler’s formula. The $(2,6)$-fullerene comes from Došlić’s $(k,6)$-fullerene, a 2-connected 3-regular plane graph with only $k$-length faces and hexagons. Došlić showed that the $(k,6)$-fullerenes only exist for $k=2$, 3, 4, or 5, and some of the structural properties of $(k,6)$-fullerene for $k=3$, 4, or 5 were studied. The structural properties, such as connectivity, extendability, resonance, and anti−Kekulé number, are very useful for studying the number of perfect matchings in a graph, and thus for the study of the stability of the molecular graphs. In this paper, we study the properties of $(2,6)$-fullerene. We discover that the edge-connectivity of $(2,6)$-fullerenes is 2. Every $(2,6)$-fullerene is 1-extendable, but not 2-extendable (F is called $n\text{-}$$extendable$ ($\left|V\right(F\left)\right|\ge 2n+2$) if any matching of n edges is contained in a perfect matching of F). F is said to be k-resonant ($k\ge 1$) if the deleting of any i ($0\le i\le k$) disjoint even faces of F results in a graph with at least one perfect matching. We have that every $(2,6)$-fullerene is 1-resonant. An edge set, S, of F is called an anti−Kekulé set if $F-S$ is connected and has no perfect matchings, where $F-S$ denotes the subgraph obtained by deleting all edges in S from F. The anti−Kekulé number of F, denoted by $ak\left(F\right)$, is the cardinality of a smallest anti−Kekulé set of F. We have that every $(2,6)$-fullerene F with $\left|V\right(F\left)\right|>6$ has anti−Kekulé number 4. Further we mainly prove that there exists a $(2,6)$-fullerene F having $f_F$ hexagonal faces, where $f_F$ is related to the two parameters n and m. Full article

Every red–blue coloring of the edges of a graph G results in a sequence $G_1$, $G_2$, …, $G_{\mathsf{\ell }}$ of pairwise edge-disjoint monochromatic subgraphs $G_i$ ($1\le i\le \mathsf{\ell }$) of size i, such that $G_i$ is isomorphic to a subgraph of $G_{i+1}$ for $1\le i\le \mathsf{\ell }-1$. Such a sequence is called a Ramsey chain in G, and $A{R}_c\left(G\right)$ is the maximum length of a Ramsey chain in G, with respect to a red–blue coloring c. The Ramsey index $AR\left(G\right)$ of G is the minimum value of $A{R}_c\left(G\right)$ among all the red–blue colorings c of G. If G has size m, then $\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)\le m<\left(\genfrac{}{}{0pt}{}{k+2}{2}\right)$ for some positive integer k. It has been shown that there are infinite classes S of graphs, such that for every graph G of size m in S, $AR\left(G\right)=k$ if and only if $\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)\le m<\left(\genfrac{}{}{0pt}{}{k+2}{2}\right)$. Two of these classes are the matchings $m{K}_2$ and paths $P_{m+1}$ of size m. These are both subclasses of linear forests (a forest of which each of the components is a path). It is shown that if F is any linear forest of size m with $\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)<m<\left(\genfrac{}{}{0pt}{}{k+2}{2}\right)$, then $AR\left(F\right)=k$. Furthermore, if F is a linear forest of size $\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)$, where $k\ge 4$, that has at most $\left(\genfrac{}{}{0pt}{}{k-1}{2}\right)$ components, then $AR\left(F\right)=k$, while for each integer t with $\left(\genfrac{}{}{0pt}{}{k-1}{2}\right)<t<\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)$ there is a linear forest F of size $\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)$ with t components, such that $AR\left(F\right)=k-1$. Full article

We study an abstract variant of squares (and shuffle squares) defined by a constraint graph G, specifying which pairs of words form a square. So, a shuffle G-square is a word that can be split into two disjoint subwords U and W (of the same length), which are joined by an edge. This setting generalizes a recently introduced model of shuffle squares based on word symmetry and permutations. By using the probabilistic method, we provide a sufficient condition for a constraint graph G guaranteeing the avoidability of shuffle G-squares. By a more-elementary method (known as Rosenfeld counting), we prove that G-squares are avoidable over an alphabet of size $4\alpha$, $\alpha >1$, provided that the degree of every word of length n in G is at most ${\alpha }^n$. We also introduce the concept of the cutting distance between words and state several conjectures involving this notion and various kinds of shuffle squares. We suspect that, for every $k⩾2$, there is a constant $c_k$ such that every even word can be turned into a shuffle square by cutting it in at most $c_k$ places and rearranging the resulting pieces. We present some computational, as well as theoretical evidence in favor of this conjecture. Full article

High availability is vital for network operators to ensure reliable services. Network faults can disrupt functionality and require quick recovery. Multipath networking enhances availability through load balancing and optimal link utilization. However, equal-cost multipath (ECMP) routing has limitations in effectively using multipaths, decreasing network availability. This paper proposes a three-phase disjoint-path framework that improves availability by directing traffic flows through separate paths. The framework provides effective load balancing and meets various service requirements. It includes the Optimization phase for identifying optimal multipath solutions, the Path Separation phase for dividing the multipath into working and backup sets, and the Quality Assessment phase for evaluating the robustness of both sets using topological metrics and micro-based characteristics. The simulations demonstrate the proposed framework’s validation and effectiveness in enhancing network availability. Full article

Federated learning (FL) has been broadly adopted in both academia and industry in recent years. As a bridge to connect the so-called “data islands”, FL has contributed greatly to promoting data utilization. In particular, FL enables disjoint entities to cooperatively train a shared model, while protecting each participant’s data privacy. However, current FL frameworks cannot offer privacy protection and reduce the computation overhead at the same time. Therefore, its implementation in practical scenarios, such as edge computing, is limited. In this paper, we propose a novel FL framework with spiking neuron models and differential privacy, which simultaneously provides theoretically guaranteed privacy protection and achieves low energy consumption. We model the local forward propagation process in a discrete way similar to nerve signal travel in the human brain. Since neurons only fire when the accumulated membrane potential exceeds a threshold, spiking neuron models require significantly lower energy compared to traditional neural networks. In addition, to protect sensitive information in model gradients, we add differently private noise in both the local training phase and server aggregation phase. Empirical evaluation results show that our proposal can effectively reduce the accuracy of membership inference attacks and property inference attacks, while maintaining a relatively low energy cost. blueFor example, the attack accuracy of a membership inference attack drops to 43% in some scenarios. As a result, our proposed FL framework can work well in large-scale cross-device learning scenarios. Full article

One of the important issues in evaluating an interconnection network is to study the hamiltonian cycle embedding problems. A graph G is spanning k-edge-cyclable if for any k independent edges $e_1,e_2,\dots ,e_k$ of G, there exist k vertex-disjoint cycles $C_1,C_2,\dots ,C_k$ in G such that $V\left(C_1\right)\cup V\left(C_2\right)\cup \cdots \cup V\left(C_k\right)=V\left(G\right)$ and $e_i\in E\left(C_i\right)$ for all $1\le i\le k$. According to the definition, the problem of finding hamiltonian cycle focuses on $k=1$. The notion of spanning edge-cyclability can be applied to the problem of identifying faulty links and other related issues in interconnection networks. In this paper, we prove that the n-dimensional hypercube $Q_n$ is spanning k-edge-cyclable for $1\le k\le n-1$ and $n\ge 2$. This is the best possible result, in the sense that the n-dimensional hypercube $Q_n$ is not spanning n-edge-cyclable. Full article