The Vertex Decomposable Property of Graphs Dual to $r$-Partite
Received:January 09, 2020  Revised:July 29, 2020
Key Word: vertex decomposable   Cohen-Macaulay   unmixedness   graphs dual to $r$-partite  
Fund ProjectL:Natural Science Foundation of Shanghai (No. 19ZR1424100); National Natural Science Foundation of China (No. 11971338)
Author NameAffiliationE-mail
Saba Yasmeen School of Mathematical Sciences, Shanghai Jiao Tong University saba@sjtu.edu.cn 
Tongsuo Wu School of Mathematical Sciences, Shanghai Jiao Tong University tswu@sjtu.edu.cn 
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Abstract:
      Let $G$ be a non-complete graph such that its complement $\ol{G}$ is $r$-partite. In this paper, properties of the graph $G$, including the unmixed property and the sequentially Cohen-Macaulay property, are studied. For $r=2,3$, some necessary and sufficient conditions are established for $G$ to be \vd \,when $G$ is dual to $r$-partite; some sufficient conditions are given for $r\ge 4$ and, connectedness of ${\rm Ind}(G)$ is also discussed.
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