A cactus graph is a connected graph in which every block is either an edge or a cycle. In this paper, we will examine cactus graphs where all the blocks are $3$-cycles, i.e., triangular cactus graphs, of diameter $4$. Our main focus is to prove that the corresponding edge ring of this family of graphs satisfies Serre's condition (S₂). We use a criterion due to Katt\"an for non-normal affine semigroup rings.
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Dinu et al. (2024) studied this question.
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