Strong Matching Preclusion for the Alternating Group Graphs and Split-Stars

Abstract

The strong matching preclusion number of a graph is the minimum number of vertices and edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. This is an extension of the matching preclusion problem and has recently been introduced by Park and Ihm.15 In this paper, we examine properties of strong matching preclusion for alternating group graphs, by finding their strong matching preclusion numbers and categorizing all optimal solutions. More importantly, we prove a general result on taking a Cartesian product of a graph with K2 (an edge) to obtain the corresponding results for split-stars.

Measuring the Vulnerability of Alternating Group Graphs and Split-Star Networks in Terms of Component Connectivity

“…In this paper, we study -connectivity of the n-dimensional alternating group graph AG n and the n-dimensional splitstars S 2 n (defined later in Section II), which were introduced by Jwo et al [33] and Cheng et al [16], respectively, for serving as interconnection network topologies of computing systems. The two families of graphs have received much attention because they have many nice properties such as vertex-transitive, strongly hierarchical, maximally connected (i.e., the connectivity is equal to its regularity), and with a small diameter and average distance.…”

“…In particular, Cheng et al [14] showed that alternating group graphs and split-stars are superior to the n-cubes and star graphs under the comparison using an advanced vulnerability measure called toughness, which was defined in [22]. For the two families of graphs, many researchers were attracted to study fault tolerant routing [12], fault tolerant embedding [5], [6], [42], matching preclusion [2], [11], restricted connectivity [15], [25], [35], [36], [48] and diagnosability [10], [25], [30], [34]- [36], [41]. Moreover, alternating group graphs are also edge-transitive and possess stronger and rich properties on Hamiltonicity (e.g., it has been shown to be not only pancyclic and Hamiltonian-connected [33] but also panconnected [6], panpositionable [40] and mutually independent Hamiltonian [39]).…”

“…Let S 2 n,E and S 2 n,O be subgraphs of S 2 n induced by the sets of even permutations and odd permutation, respectively, in which the adjacency applied to each subgraph is precisely using the edge of 3-rotation. Clearly, S 2 n,E is the alternating group graph AG n , and An independent set of a graph G is a subset S ⊆ V (G) such that any two vertices of S are nonadjacent in G. For u ∈ V (G), we define N G (u) = {v ∈ V (G) : (u, v) ∈ E(G)}, i.e., the set of neighbors of u. Moreover, for S ⊆ V (G), we define When the graph G is clear from the context, the subscript in the above notations are omitted.…”

Conditional Strong Matching Preclusion of the Alternating Group Graph

“…Proposition 2.1. [4] Let G be a r-regular even graph with r ≥ 2. Suppose that smp(G) = r. Then every basic optimal strong matching preclusion set is trivial.…”

Strong Matching Preclusion for Augmented Butterfly Networks

“…Park and Ihm [8] also studied the problem of strong matching preclusion under the condition that no isolated vertex is created as a result of faults, and established the conditional strong matching preclusion number for the class of restricted hypercube-like graphs, which include most non bipartite hypercube-like networks found in the literature. SMP numbers of augmented cubes, arrangement graphs, alternating group graphs and split-star, pancake graphs, 2 -matching composition networks, k -ary n cubes, n-dimensional torus networks, k-composition networks are also investigated; see [1,[9][10][11][12][13][14].…”