The Topological Connectivity of the Independence Complex of Circular-Arc Graphs
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...
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| Format: | Article |
| Language: | English |
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Emrah Evren KARA
2019-12-01
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| Series: | Universal Journal of Mathematics and Applications |
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| Online Access: | https://dergipark.org.tr/tr/download/article-file/901755 |
| _version_ | 1797350232106729472 |
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| author | Yousef Abd Algani |
| author_facet | Yousef Abd Algani |
| author_sort | Yousef Abd Algani |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-08T12:41:43Z |
| format | Article |
| id | doaj.art-c9db143b7b104f68a5f692528f57be47 |
| institution | Directory Open Access Journal |
| issn | 2619-9653 |
| language | English |
| last_indexed | 2024-03-08T12:41:43Z |
| publishDate | 2019-12-01 |
| publisher | Emrah Evren KARA |
| record_format | Article |
| series | Universal Journal of Mathematics and Applications |
| spelling | doaj.art-c9db143b7b104f68a5f692528f57be472024-01-21T10:12:36ZengEmrah Evren KARAUniversal Journal of Mathematics and Applications2619-96532019-12-012415916910.32323/ujma.5564571225The Topological Connectivity of the Independence Complex of Circular-Arc GraphsYousef Abd AlganiLet 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.https://dergipark.org.tr/tr/download/article-file/901755topological connectivityindependence complexcircular-arc graphs |
| spellingShingle | Yousef Abd Algani The Topological Connectivity of the Independence Complex of Circular-Arc Graphs Universal Journal of Mathematics and Applications topological connectivity independence complex circular-arc graphs |
| title | The Topological Connectivity of the Independence Complex of Circular-Arc Graphs |
| title_full | The Topological Connectivity of the Independence Complex of Circular-Arc Graphs |
| title_fullStr | The Topological Connectivity of the Independence Complex of Circular-Arc Graphs |
| title_full_unstemmed | The Topological Connectivity of the Independence Complex of Circular-Arc Graphs |
| title_short | The Topological Connectivity of the Independence Complex of Circular-Arc Graphs |
| title_sort | topological connectivity of the independence complex of circular arc graphs |
| topic | topological connectivity independence complex circular-arc graphs |
| url | https://dergipark.org.tr/tr/download/article-file/901755 |
| work_keys_str_mv | AT yousefabdalgani thetopologicalconnectivityoftheindependencecomplexofcirculararcgraphs AT yousefabdalgani topologicalconnectivityoftheindependencecomplexofcirculararcgraphs |