A New Construction for Cohen–Macaulay Graphs

Let G be a finite simple graph on a vertex set V(G) = {x 11,…, x n1}. Also let m 1,…, m n ≥ 2 be integers and G 1,…, G n be connected simple graphs on the vertex sets V(G i ) = {x i1,…, x im i }. In this article, we provide necessary and sufficient conditions on G 1,…, G n for which the graph obtained by attaching the 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 arbitrary graphs.