A disjoint path problem in the alternating group graph (original) (raw)

Minimum Neighborhood of Alternating Group Graphs

IEEE Access, 2019

The minimum neighborhood and combinatorial property are two important indicators of fault tolerance of a multiprocessor system. Given a graph G, θ G (q) is the minimum number of vertices adjacent to a set of q vertices of G (1 ≤ q ≤ |V (G)|). It is meant to determine θ G (q), the minimum neighborhood problem (MNP). In this paper, we obtain θ AG n (q) for an independent set with size q in an n-dimensional alternating group graph AG n , a well-known interconnection network for multiprocessor systems. We first propose some combinatorial properties of AG n. Then, we study the MNP for an independent set of two vertices and obtain that θ AG n (2) = 4n − 10. Next, we prove that θ AG n (3) = 6n − 16. Finally, we propose that θ AG n (4) = 8n − 24. INDEX TERMS Minimum neighborhood, combinatorial property, fault tolerance, independent set, alternating group graphs.

Structure and substructure connectivity of alternating group graphs

Applied Mathematics and Computation, 2021

The connectivity is an important indicator to evaluate the robustness of a network. Many works have focused on connectivity-based reliability analysis for decades. As a generalization of connectivity, H-structure connectivity and H-substructure connectivity were proposed to evaluate the robustness of networks. In this paper, we investigate the H-structure connectivity and H-substructure connectivity of alternating group graph AG n when H is isomorphic to K 1, t , P l and C k , which are generalizations of the previous results for

Edge-connectivity in -path graphs

Discrete Mathematics, 2004

The P k (G)-path graph corresponding to a graph G has for vertices the set of all paths of length k in G. Two vertices are joined by an edge if and only if the intersection of the corresponding paths forms a path of length k − 1 in G, and their union forms either a cycle or a path of length k + 1. Path graphs were introduced by Broersma and Hoede (J. Graph. Theory 13 as a generalization of line graphs, because for k = 1, path graphs are just line graphs. Results on the edge-connectivity of line graphs are given by Chartrand and Stewart (12). The connectivity of P k -path graphs has been studied by Knor and Niepel (Graph Theory 20 (2000) 181), where they proved a necessary and sufficient condition for the P k (G)-path graphs to be disconnected, assuming that G has girth of at least k + 1. Going one step further, we prove in this work that the edge-connectivity of P k (G) is at least (P k (G)) (G) − 1 for a graph G of girth at least k + 1 and minimum degree (G) 2. Furthermore, we show (P k (G)) 2 (G) − 2 provided that (G) 3.

Note on the problem of partially link disjoint paths

This paper discusses the problem of partially link disjoint paths in communication networks. The problem is situated in between the well-known shortest path generation and the synthesis of two disjoint paths. The contributions of this paper are twofold: firstly, topologies and relevant graph properties are investigated and the degree of divergence is defined as a metric to distinguish the degree of commonality for the two partially disjoint paths. Secondly, heuristic approaches are introduced that solve the problem of generating partly disjoint paths.

On the disjoint paths problem

Operations Research Letters, 2007

Using flow and matching algorithms to solve the problem of finding disjoint paths through a given node, and with a technique of Chekuri and Khanna, we give an O( √ n) approximation for the edge-disjoint paths problem in undirected graphs, directed acyclic graphs and directed graphs with edge capacity at least 2.

2-CONNECTIVITY of the Power Graph of Finite Alternating Groups

For a finite group G, let P0(G), O0(G) and P0(T (G)) be the proper power graph, the proper order graph and the proper power type graph of G respectively. In this paper we find the number of connected components of those graphs when G = An. As a consequence we deduce that the power graph P (An) is 2-connected if and only if n = 3 or n, n−1, n−2, n 2 and n−1 2 are not primes. This gives an infinite family of counterexamples to a conjecture posed by Pourgholi et al. in [3].

On the total-diameter of connection networks

Theoretical Computer Science, 2000

We study connection networks in which certain pairs of nodes have to be connected by k edge-disjoint paths, and study bounds for the minimal sum of lengths of such k paths. We define the related notions of total k -distance for a pair of nodes and total k -diameter of a connection network, and study the value TD k (d) which is the maximal such total k -diameter of a network with diameter d. These notions have applications in fault-tolerant routing problems, in ATM networks, and in compact routing in networks. We prove an upper bound on TD k (d) and a lower bound on the growth of TD k (d) as functions of k and d; those bounds are tight, θ(d k ), when k is fixed. Specifically, we prove that TD k (d) ≤ 2 k−1 d k , with the exceptions TD 2 (1) = 3, TD 3 (1) = 5, and that for every k, d 0 > 0, there exists (a) an integer d ≥ d 0 such that TD k (d) ≥ d k /k k , and (b) a k-connected simple graph G with diameter d such that d ≥ d 0 , and whose total k -diameter is at least (d − 2) k /k k .

Paths and circuits in 𝔾-graphs

Involve, a Journal of Mathematics, 2008

For a group G with generating set S = {s 1 , s 2 ,. .. , s k }, the ‫-އ‬graph of G, denoted (G, S), is the graph whose vertices are distinct cosets of s i in G. Two distinct vertices are joined by an edge when the set intersection of the cosets is nonempty. In this paper, we study the existence of Hamiltonian and Eulerian paths and circuits in (G, S). MSC2000: 05C25, 20F05.

A note on an extremal problem for group-connectivity

European Journal of Combinatorics, 2014

In , an extremal graph theory problem is proposed for group connectivity: for an abelian group A with |A| ≥ 3 and an integer n ≥ 3, find ex(n, A), where ex(n, A) is the maximum number so that every n-vertex simple graph with at most ex(n, A) edges is not A-connected. In this paper, we determine the values ex(n, A) for A = Z k where k ≥ 3 is an odd integer and for A = Z 4 , each of which solves some open problem proposed in . Furthermore, the values ex(n, Z 4 ) also imply a characterization of Z 4 -connected graphic sequences.