@article{article_556457, title={The Topological Connectivity of the Independence Complex of Circular-Arc Graphs}, journal={Universal Journal of Mathematics and Applications}, volume={2}, pages={159–169}, year={2019}, DOI={10.32323/ujma.556457}, author={Abd Algani, Yousef}, keywords={Topological connectivity,Independence Complex,Circular-Arc graphs}, abstract={<div style="text-align:justify;"> <span style="font-size:14px;">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. </span> <br /> </div>}, number={4}, publisher={Emrah Evren KARA}