The Topological Connectivity of the Bipartite Graph’s Independence Complex By Yousef Methkal Abd Algani and Amal Sharif-Rasslan
Let (∁) signify a simplicial complex; the topological connectivity of ∁ is denoted by ζ (∁), which equals ∁+2, where ζ is a function from the class of graphs to the set ℤ+=012…∞. Let the function ζ possesses the following characteristics: 1. ζK0=0. 2. For any graph GVE, there exists an edge xyin G such that ζGVE−xy≥ζGVE where (GVE−xy) means to remove the edge xy from the graph GVE), and ζGVE−Δxy≥ζGVE−1where (GVE−Δ(xy) means to remove the vertices xandy and all their neighbors. Then μ(ΓGVE≥ζGVE. Let ζ0 denote the maximal function ζ that satisfies the aforementioned constraints. Berger proved that μ(ΓGVE≥ζ0GVE for trees and for complements of chordal graphs. Kawamura proved the same theorem for chordal graphs, and Abd Algani proved it for circular-arc graph: Let GVE be a circular-arc graph; if ζ0GVE≤2,thenμΓGVE≤2. In this chapter, we will prove the following theorem: Let GVE be a bipartite graph; if ζ0GVE≤2,thenμΓGVE≤2.
Part of the book: Topology
