Minimum Neighborhood of Alternating Group Graphs

Abstract

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.

Component Reliability Evaluation on Split-Stars

“…• four components, the small three being all a vertex, respectively. Proposition 1 [18]: Let {u, v} be an independent set of two vertices on a 4-cycle of an (n − 1)-subgraph of AG n . Then the four external neighbors of u, v are in four different subgraphs.…”

“…Proposition 2 [18]: Let {u, v} be an independent set of two isolated vertices on a 2-path of an (n−1)-subgraph of AG n such that uv / ∈ E(AG n ). If |N (u) ∩ N (v)| = 1, then the four external neighbors of u, v are in three different subgraphs.…”

“…If |N (u) ∩ N (v)| = 1, then the four external neighbors of u, v are in three different subgraphs. Proposition 3 [18]: Let D be an independent set in V (AG n ) (n ≥ 4). We have the following results:…”