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

Authors : Yousef ABD ALGANİ

Pages : 159-169

Doi:10.32323/ujma.556457

View : 20 | Download : 11

Publication Date : 2019-12-26

Article Type : Research Paper

Abstract :Let us denoted the topological connectivity of a simplicial complex $C$ plus 2 by $\etainsert ignore into journalissuearticles values(C);$. 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 $\psi{insert ignore into journalissuearticles values(K_{0});}$=0. \newline For every graph G there exists an edge $e=insert ignore into journalissuearticles values(x,y);$ of $G$ such that $$\psi{insert ignore into journalissuearticles values(G-e);}\geq{\psi{insert ignore into journalissuearticles values(G);}}$$ insert ignore into journalissuearticles values(where $G-e$ is obtained from $G$ by the removal of the edge $e$);, and $$\psi{insert ignore into journalissuearticles values(G-Ninsert ignore into journalissuearticles values(\lbrace x,y \rbrace););}\geq{\psi{insert ignore into journalissuearticles values(G);}}-1$$  then $$\eta{insert ignore into journalissuearticles values(\mathcal{I}{insert ignore into journalissuearticles values(G);});}\geq\psi{insert ignore into journalissuearticles values(G);}$$ insert ignore into journalissuearticles values(where $insert ignore into journalissuearticles values(G-Ninsert ignore into journalissuearticles values(\lbrace x,y \rbrace););$ is obtained from $G$ by the removal of  all neighbors of $x$ and $y$ insert ignore into journalissuearticles values(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: $$\eta{insert ignore into journalissuearticles values(\mathcal{I}{insert ignore into journalissuearticles values(G);});}=\psi_{0}{insert ignore into journalissuearticles values(G);}$$ 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 $\psi_0insert ignore into journalissuearticles values(G);\leq 2$ then $\etainsert ignore into journalissuearticles values(\mathcal{I}insert ignore into journalissuearticles values(G););\leq 2$. Prior the attempt to verify the previously mentioned cases, we need a few preparations which will be discussed in the introduction.
Keywords : Topological connectivity, Independence Complex, Circular Arc graphs