Cyclic Vertex (Edge) Connectivity of Burnt Pancake Graphs

Edge and vertex connectivity, as well as edge and vertex cuts are among the most basic and fundamental concepts in graph theory. In particular, they are naturally significant in a networking context as they are a measure for the rate at which information can be transferred across a network. While in a traditional, sequential setting, there is a rich literature (in particular on problems related to edge connectivity and edge cuts), until recently, much less was known from a distributed algorithms point of view. In my talk, I will discuss and give partial answers to some of the following basic questions. Using distributed algorithms, how fast can we compute or approximate the edge or vertex connectivity and is it possible to efficiently find small cuts in a network? Assuming, we have a network with good connectivity properties, to what extent is it possible to exploit this in order to speed up distributed computations? Where such properties can be exploited, are there network structures that allow to make use of good connectivity in a structured (and somewhat canonical) way and can we construct such structures in a distributed manner?

Let p p be an odd prime. Let P P be a finite p p -group of class 2 2 and exponent p p , whose commutator quotient P / [ P , P ] P/[P,P] is of order p n p^n . We define two parameters for P P related to central decompositions. The first parameter, κ ( P ) \kappa (P) , is the smallest integer s s for the existence of a subgroup S S of P P satisfying (1) S ∩ [ P , P ] = [ S , S ] S\cap [P,P]=[S,S] , (2) | S / [ S , S ] | = p n − s |S/[S,S]|=p^{n-s} , and (3) S S is centrally decomposable. The second parameter, λ ( P ) \lambda (P) , is the smallest integer s s for the existence of a central subgroup N N of order p s p^s , such that P / N P/N is centrally decomposable. While defined in purely group-theoretic terms, these two parameters generalise, respectively, the vertex and edge connectivities of graphs: For a simple undirected graph G G , through the classical procedures of Baer (Trans. Amer. Math. Soc., 1938), Tutte (J. Lond. Math. Soc., 1947) and Lovász (B. Braz. Math. Soc., 1989), there is a p p -group P G P_G of class 2 2 and exponent p p that is naturally associated with G G . Our main results show that the vertex connectivity κ ( G ) \kappa (G) equals κ ( P G ) \kappa (P_G) , and the edge connectivity λ ( G ) \lambda (G) equals λ ( P G ) \lambda (P_G) . We also discuss the relation between κ ( P ) \kappa (P) and λ ( P ) \lambda (P) for a general p p -group P P of class 2 2 and exponent p p , as well as the computational aspects of these parameters. In particular, our main results imply that the p p -group central decomposition algorithm of Wilson (J. Algebra and J. of Group Theory, 2009) can be used to solve the graph connectivity problem.

Cubic fuzzy graphs (CFGs) offer greater utility as compared to interval-valued fuzzy graphs and fuzzy graphs due to their ability to represent the degree of membership for vertices and edges using both interval and fuzzy number forms. The significance of these concepts motivates us to analyze and interpret intricate networks, enabling more effective decision making and optimization in various domains, including transportation, social networks, trade networks, and communication systems. This paper introduces the concepts of vertex and edge connectivity in CFGs, along with discussions on partial cubic fuzzy cut nodes and partial cubic fuzzy edge cuts, and presents several related results with the help of some examples to enhance understanding. In addition, this paper introduces the idea of partial cubic α-strong and partial cubic δ-weak edges. An example is discussed to explain the motivation behind partial cubic α-strong edges. Moreover, it delves into the introduction of generalized vertex and edge connectivity in CFGs, along with generalized partial cubic fuzzy cut nodes and generalized partial cubic fuzzy edge cuts. Relevant results pertaining to these concepts are also discussed. As an application, the concept of generalized partial cubic fuzzy edge cuts is applied to identify regions that are most affected by trade deficits resulting from street crimes. Finally, the research findings are compared with the existing method to demonstrate their suitability and creativity.

The decomposition of a connected graph by the set of its cut-vertices, sometimes called the \\"block decomposition\\" or \\"block tree\\" of a graph, is a well known and basic concept in graph theory. This decomposition, however, does not provide meaningful information when applied to a k-connected graph for k > 1. There has been a number of attempts to generalize the construction of the block decomposition of a graph for the case of k-connected graphs. Notably, Tutte constructed a tree that describes the mutual arrangement of 2-cutsets in a 2-connected graph. This decomposition has some similarities to the block decomposition of a connected graph. In other works, a block of a k-connected graph was defined as a maximal (k+1)-connected subgraph. Karpov described the decomposition of a k-connected graph by the set of k-cutsets that are not separated by any other k-cutset of the graph. Karpov also described some special properties of his decomposition for the case of a 2-connected graph. The decompositions defined by Karpov and Tutte for the case of a 2-connected graph share some similarities. In this work, we present a self-contained description of Karpov\\'s decomposition. We also present some applications to the study of planarity, the chromatic number, critically 2-connected graphs, and the partition of certain 2-connected graphs into three connected subgraphs.

The vertex connectivity of an m-edge n-vertex undirected graph is the smallest number of vertices whose removal disconnects the graph, or leaves only a singleton vertex. In this paper, we give a reduction from the vertex connectivity problem to a set of maxflow instances. Using this reduction, we can solve vertex connectivity in (mα) time for any α ≥ 1, if there is a mα-time maxflow algorithm. Using the current best maxflow algorithm that runs in m4/3+o(1) time (Kathuria, Liu and Sidford, FOCS 2020), this yields a m4/3+o(1)-time vertex connectivity algorithm. This is the first improvement in the running time of the vertex connectivity problem in over 20 years, the previous best being an O(mn)-time algorithm due to Henzinger, Rao, and Gabow (FOCS 1996). Indeed, no algorithm with an o(mn) running time was known before our work, even if we assume an (m)-time maxflow algorithm. Our new technique is robust enough to also improve the best O(mn)-time bound for directed vertex connectivity to mn1−1/12+o(1) time

Brain connectivity studies often refer to brain areas as graph nodes and connections between nodes as edges, and aim to identify neuropsychiatric phenotype-related connectivity patterns. When performing group-level brain connectivity alternation analyses, it is critical to model the dependence structure between multivariate connectivity edges to achieve accurate and efficient estimates of model parameters. However, specifying and estimating dependencies between connectivity edges presents formidable challenges because (i) the dimensionality of parameters in the covariance matrix is high (of the order of the fourth power of the number of nodes); (ii) the covariance between a pair of edges involves four nodes with spatial location information; and (iii) the dependence structure between edges can be related to unknown network topological structures. Existing methods for large covariance/precision matrix regularization and spatial closeness-based dependence structure specification/estimation models may not fully address the complexity and challenges. We develop a new Bayesian nonparametric model that unifies information from brain network areas (nodes), connectivity (edges), and covariance between edges by constructing the function of covariance matrix based on the underlying network topological structure. We perform parameter estimation using an efficient Markov chain Monte Carlo algorithm. We apply our method to resting-state functional magnetic resonance imaging data from 60 subjects of a schizophrenia study and simulated data to demonstrate the performance of our method.

The vertex connectivity of a graph is the minimum number of vertices whose removal disconnects the graph, and it is a fundamental notion that measures the level of interconnectedness of vertices. Mining subgraphs with high vertex connectivity has wide applications in robust and efficient communication networks and the organization of task groups. However, existing approaches often neglect size constraints, leading to subgraphs that are either too large or too small for practical use. Therefore, in this paper, we introduce the size constraint and formulate the graph downsizing problem: Given a graph G with n vertices and a positive integer $$\tau &lt;n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>τ</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> , find an induced subgraph of size $$n-\tau $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mi>τ</mml:mi> </mml:mrow> </mml:math> that maximizes vertex connectivity. We first analyze the computational complexity of the graph downsizing problem and show that the problem is hard in several aspects. To solve the problem in practice, we use a nesting tree structure to represent relations between maximal cohesive subgraphs in G . This structure allows us to develop an efficient heuristic, called inundation downsizing algorithm, to find highly vertex-connected subgraphs of different sizes. Experimental results reveal that our heuristic is able to produce almost optimal solutions on real-world datasets with up to 1 million vertices while incurring small time costs.

Distance spectral radius of digraphs with given connectivity

A fundamental result by Karger [10] states that for any λ-edge-connected graph with n nodes, independently sampling each edge with probability p = Ω(log n/λ) results in a graph that has edge connectivity Ω(λp), with high probability. This paper proves the analogous result for vertex connectivity, when sampling vertices. We show that for any k-vertex-connected graph G with n nodes, if each node is independently sampled with probability p = Ω([EQUATION]log n/k), then the subgraph induced by the sampled nodes has vertex connectivity Ω(kp2), with high probability. This bound improves upon the recent results of Censor-Hillel et al. [6], and is existentially optimal.

Extra edge connectivity and diagnosability have been employed to investigate the fault tolerance properties of network structures. The p-extra edge connectivity λp(Γ) of a graph Γ was introduced by Fàbrega and Fiol in 1996. In this paper, we find the exact values of p-extra edge connectivity of some special graphs. Moreover, we give some upper and lower bounds for λp(Γ), and graphs with λp(Γ)=1,2,n2n2−1,n2n2 are characterized. Finally, we obtain the three extremal results for the p-extra edge connectivity.

Connectivity keeping caterpillars and spiders in 2-connected graphs

Parity of Cycles Containing Specified Edges

It is shown that the vertex connectivity of the block-intersection graph of a balanced incomplete block design,BIBD (v, k, 1), is equal to its minimum degree. A similar statement is proved for the edge connectivity of the block-intersection graph of a pairwise balanced design,PBD (v, K, 1). A partial result on the vertex connectivity of these graphs is also given. Minimal vertex and edge cuts for the corresponding graphs are characterized.

Given an undirected graph G and a positive integer k, the k-vertex-connectivity augmentation problem is to find a smallest set F of new edges for which G+F is k-vertex-connected. Polynomial algorithms for this problem have been found only for k ≤ 4 and a major open question in graph connectivity is whether this problem is solvable in polynomial time in general.In this paper we develop the first algorithm which delivers an optimal solution in polynomial time for every fixed k. In the case when the size of an optimal solution is large compared to k, we also give a min-max formula for the size of a smallest augmenting set.A key step in our proofs is a complete solution of the augmentation problem for a new family of graphs which we call k-independence free graphs. We also prove new splitting off theorems for vertex connectivity.

We first consider the problem of partitioning the edges of a graph G into bipartite cliques such the total order of the cliques is minimized, where the order of a clique is the number of vertices in it. It is shown that the problem is NP-complete. We then prove the existence of a partition of small total order in a sufficiently dense graph and devise an efficient algorithm to compute such a partition and the running time. Next, we define the notion of a compression of a graph G and use the result on graph partitioning to efficiently compute an optimal compression for graphs of a given size. An interesting application of the graph compression result arises from the fact that several graph algorithms can be adapted to work with the compressed representation of the input graph, thereby improving the bound on their running times, particularly on dense graphs. This makes use of the trade-off result we obtain from our partitioning algorithm. The algorithms analyzed include those for matchings, vertex connectivity, edge connectivity, and shortest paths. In each case, we improve upon the running times of the best-known algorithms for these problems.