The Topological Connectivity of the Independence Complex of Circular-Arc Graphs

Abstract

Let us denoted the topological connectivity of a simplicial complex $C$ plus 2 by $\eta(C)$. Let $\psi$ be a function from class of graphs to the set of positive integers together with $\infty$. Suppose $\psi$ satisfies the following properties: \newline $\psi{(K_{0})}$=0. \newline For every graph G there exists an edge $e=(x,y)$ of $G$ such that $$\psi{(G-e)}\geq{\psi{(G)}}$$ (where $G-e$ is obtained from $G$ by the removal of the edge $e$), and $$\psi{(G-N(\lbrace x,y \rbrace))}\geq{\psi{(G)}}-1$$ then $$\eta{(\mathcal{I}{(G)})}\geq\psi{(G)}$$ (where $(G-N(\lbrace x,y \rbrace))$ is obtained from $G$ by the removal of all neighbors of $x$ and $y$ (including, of course, $x$ and $y$ themselves). Let us denoted the maximal function satisfying the conditions above by $\psi_0$. Berger [3] prove the following conjecture: $$\eta{(\mathcal{I}{(G)})}=\psi_{0}{(G)}$$ for trees and completements of chordal graphs. Kawamura [2] proved conjecture, for chordal graphs. Berger [3] proved Conjecture for trees and completements of chordal graphs. In this article I proved the following theorem: Let $G$ be a circular-arc graph $G$ if $\psi_0(G)\leq 2$ then $\eta(\mathcal{I}(G))\leq 2$. Prior the attempt to verify the previously mentioned cases, we need a few preparations which will be discussed in the introduction.