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A new construction for Cohen-Macaulay graphs

Let $G$ be a finite simple graph on a vertex set $V(G)=\{x_{11}, \ldots, x_{n1}\}$. Also let $m_1, \ldots,m_n \geq 2$ be integers and $G_1, \ldots, G_n$ be connected simple graphs on the vertex sets $V(G_i)=\{x_{i1}, \ldots, x_{im_i}\}$. In this paper, we provide necessary and sufficient conditions on $G_1, \ldots, G_n$ for which the graph obtained by attaching $G_i$ to $G$ is unmixed or vertex decomposable. Then we characterize Cohen--Macaulay and sequentially Cohen--Macaulay graphs obtained by attaching the cycle graphs or connected chordal graphs to an arbitrary graphs.