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Let $G=(V,E)$ be a graph. If $G$ is a Konig graph or $G$ is a graph without 3-cycles and 5-cycle, we prove that the following conditions are equivalent: $Delta_{G}$ is pure shellable, $R/I_{Delta}$ is Cohen-Macaulay, $G$ is unmixed vertex decomposable graph and $G$ is well-covered with a perfect matching of Konig type $e_{1},...,e_{g}$ without square with two $e_i$s. We characterize well-covered graphs without 3-cycles, 5-cycles and 7-cycles. Also, we study when graphs without 3-cycles and 5-cycles are vertex decomposable or shellable. Furthermore, we give some properties and relations between critical, extendables and shedding vertices. Finally, we characterize unicyclic graphs with each one of the following properties: unmixed, vertex decomposable, shellable and Cohen-Macaulay.
We characterize unmixed and Cohen-Macaulay edge-weighted edge ideals of very well-covered graphs. We also provide examples of oriented graphs which have unmixed and non-Cohen-Macaulay vertex-weighted edge ideals, while the edge ideal of their underly
Given a simple undirected graph $G$ there is a simplicial complex $mathrm{Ind}(G)$, called the independence complex, whose faces correspond to the independent sets of $G$. This is a well studied concept because it provides a fertile ground for intera
We say that a pure $d$-dimensional simplicial complex $Delta$ on $n$ vertices is shelling completable if $Delta$ can be realized as the initial sequence of some shelling of $Delta_{n-1}^{(d)}$, the $d$-skeleton of the $(n-1)$-dimensional simplex. A w
In this article, we provide a complete list of simple Cohen-Macaulay codimension 2 singularities together with a list of adjacencies which is complete in the case of fat point and space curve singularities.
We study Cohen-Macaulay Hopf monoids in the category of species. The goal is to apply techniques from topological combinatorics to the study of polynomial invariants arising from combinatorial Hopf algebras. Given a polynomial invariant arising from