Two kinds of generalized connectivity of dual cubes

Abstract

Let S ⊆ V (G) and κ G (S) denote the maximum number k of edge-disjoint trees T 1 , T 2 , · · · , T k in G such that V (T i ) V (T j ) = S for any i, j ∈ {1, 2, · · · , k} and i = j. For an integer r with 2 ≤ r ≤ n, the generalized r-connectivity of a graph G is defined as κ r (G) = min{κ G (S)|S ⊆ V (G) and |S| = r}. The r-component connectivity cκ r (G) of a non-complete graph G is the minimum number of vertices whose deletion results in a graph with at least r components. These two parameters are both generalizations of traditional connectivity. Except hypercubes and complete bipartite graphs, almost all known κ r (G) are about r = 3. In this paper, we focus on κ 4 (D n ) of dual cube D n . We first show that κ 4 (D n ) = n − 1 for n ≥ 4. As a corollary, we obtain κ 3 (D n ) = n − 1 for n ≥ 4. Furthermore, we show that cκ r+1 (D n ) = rn − r(r+1) 2 + 1 for n ≥ 2 and 1 ≤ r ≤ n − 1.

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

“…So far, the exact values of -connectivity are known only for a few classes of networks and small 's. For example, -connectivity is determined on hypercube Q n for ∈ [2, n + 1] (see [32]) and ∈ [n + 2, 2n − 4] (see [51]), folded hypercube FQ n for ∈ [2, n + 2] (see [50]), dual cube D n for ∈ [2, n] (see [49]), hierarchical cubic network HCN (n) for ∈ [2, n + 1] (see [19]), complete cubic network CCN (n) for ∈ [2, n + 1] (see [20]), and generalized exchanged hypercube GEH (s, t) for 1 s t and ∈ [2, s + 1] (see [21]). Note that the number of vertices of graphs in the above classes is an exponent related to n. Also, it has been pointed out in [32] that determining -connectivity is still unsolved for most interconnection networks such as star graphs S n and alternating group graphs AG n .…”

Constructing edge-disjoint Steiner trees in Cartesian product networks

“…A lot of attention has been devoted to fault-tolerant broadcasting in networks [6,11,13,26]. In order to measure the ability of fault-tolerance, the path structure connecting two nodes must be generalized into some tree structures connecting more than two nodes, see [14,25,33,34,15,32,18,24,17].…”

Component (edge) connectivity of pancake graphs

“…The two parameters have been investigated in several interconnection networks. See for example [3,7,12,16,23,27,28,29]. Recently, the relationship between extra connectivity and component connectivity of general networks has been investigated by Li et al [14], while the relationship between extra edge connectivity and component edge connectivity of regular networks has been suggested by Hao et al.…”