Vertex decomposability and regularity of very well-covered graphs

By M. Mahmoudi, A. Mousivand, Marilena Crupi, Giancarlo Rinaldo, Naoki Terai, Siamak Yassemi. 2010

86 citations recorded by the source corpus.

A graph $G$ is well-covered if it has no isolated vertices and all the maximal independent sets have the same cardinality. If furthermore two times this cardinality is equal to $|V(G)|$, the graph $G$ is called very well-covered. The class of very well-covered graphs contains bipartite well-covered graphs. Recently in \cite{CRT} it is shown that a very well-covered graph $G$ is Cohen-Macaulay if and only if it is pure shellable. In this article we improve this result by showing that $G$ is Cohen-Macaulay if and only if it is pure vertex decomposable. In addition, if $I(G)$ denotes the edge ideal of $G$, we show that the Castelnuovo-Mumford regularity of $R/I(G)$ is equal to the maximum number of pairwise 3-disjoint edges of $G$. This improves Kummini's result on unmixed bipartite graphs.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Pharmacology

View Paper: https://www.semanticscholar.org/paper/8ac1be4eb68c89b5d5de14576f946db833cd2816

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Citations in the corpus

Matchings, coverings, and Castelnuovo-Mumford regularity (2010, result): “We recover and extend results of [19, 25], but show that the results of [27, 34] cannot be proved using this technique.”
Regularity of powers of bipartite graphs (2016, unknown): “In particular, there has been active research on bounding the homological invariants of edge ideals in terms of the combinatorial invariants of the associated graphs, see for example [3,10,12,15,17,19,22,23,26,28,32,33].”
Non-vanishingness of Betti numbers of edge ideals (2011, background): “ Also, the (Castelnuovo–Mumford) regularity of S/I(G) is deﬁned by regS/I(G) := max{j − i : βi,j(S/I(G)) 6= 0 }. In this paper, we focus on these invariants, which have studied by many authors, e.g., [2, 7, 9, 10, 11, 12, 14, 16]. In particular, Zheng [16] characterized the projective dimension and ”
Regularity of Squarefree Monomial Ideals (2013, background): “Then reg(G) = ν(G) + 1 in the following cases: (1) G is a sequentially Cohen-Macaulay bipartite graph (see [48]); (2) G is an unmixed bipartite graph (see [36]); (3) G is a very well-covered graph (see [38]); (4) G is a C5-free vertex decomposable graph (see [34], the case where G is also C4-free wa”
Symbolic powers of cover ideal of very well-covered and bipartite graphs (2016, unknown): “This class of graphs is studied from algebraic point of view in [7], [20] [21].”
Algebraic study on Cameron-Walker graphs (2013, background): “, chordal graphs, unmixed bipartite graphs and sequentially Cohen–Macaulay bipartite graphs ([9, 12, 15]; see also [7, 13, 19, 20]).”
Regularity of Edge Ideals and Their Powers (2017, background): “(see [29]); (b) Gis a weakly chordal graph (see [64]); (c) Gis sequentially Cohen-Macaulay bipartite graph (see [62]); (d) Gis unmixed bipartite graph (see [43]); (e) Gis very well-covered graph (see [46]); (f) Gis vertex decomposable graph and has no closed circuit of length 5 (see [40]); (g) Gis a”