Vertex decomposability and regularity of very well-covered graphs

Mahmoudi, M. G.; Mousivand, Amir; Crupi, Marilena; Rinaldo, Giancarlo; Terai, Naoki; Yassemi, Siamak

doi:10.1016/j.jpaa.2011.02.005

Cited by 64 publications

(63 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In particular, Corollary 2.2 implies: 3 Very well-covered graphs are localizable A well-covered graph G is said to be very well-covered if it has no isolated vertices and α(G) = |V (G)|/2. Very well-covered graphs were studied in commutative algebra, in the context of connections between properties of a graph G with properties of the simplicial complex whose faces are the independent sets of G, and of the edge ideal of G, see, e.g., [16,57]. Very well-covered graphs were characterized by Staples in 1975 [71,Theorem 1.11] and independently by Favaron in 1982 [27], as follows.…”

Section: Equivalent Formulations Of Localizabilitymentioning

confidence: 99%

Graphs vertex-partitionable into strong cliques

Hujdurović

Milanič

Ries

2018

Discrete Mathematics

View full text Add to dashboard Cite

A graph is said to be well-covered if all its maximal independent sets are of the same size. In 1999, Yamashita and Kameda introduced a subclass of well-covered graphs, called localizable graphs and defined as graphs having a partition of the vertex set into strong cliques, where a clique in a graph is strong if it intersects all maximal independent sets. Yamashita and Kameda observed that all well-covered trees are localizable, pointed out that the converse inclusion fails in general, and asked for a characterization of localizable graphs.In this paper we obtain several structural and algorithmic results about localizable graphs. Our results include a proof of the fact that every very well-covered graph is localizable and characterizations of localizable graphs within the classes of line graphs, triangle-free graphs, C 4 -free graphs, and cubic graphs, each leading to a polynomial time recognition algorithm. On the negative side, we prove NP-hardness of recognizing localizable graphs within the classes of weakly chordal graphs, complements of line graphs, and graphs of independence number three. Furthermore, using localizable graphs we disprove a conjecture due to Zaare-Nahandi about kpartite well-covered graphs having all maximal cliques of size k. Our results unify and generalize several results from the literature.

show abstract

Section: Equivalent Formulations Of Localizabilitymentioning

confidence: 99%

Graphs vertex-partitionable into strong cliques

Hujdurović

Milanič

Ries

2018

Discrete Mathematics

View full text Add to dashboard Cite

show abstract

“…[24]) that 2 ht I(G) ≥ |X(G)|. A graph G is called very well-covered (see [38]) if G is unmixed, has no isolated vertices, and 2 ht I(G) = |X(G)|.…”

Section: Small Regularity and Computing Regularitymentioning

confidence: 99%

“…(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 C 5 -free vertex decomposable graph (see [34], the case where G is also C 4 -free was proved in [5]). …”

Section: Small Regularity and Computing Regularitymentioning

confidence: 99%

Regularity of Squarefree Monomial Ideals

Harbourne

2014

Springer Proceedings in Mathematics &Amp; Statistics

View full text Add to dashboard Cite

Abstract. We survey a number of recent studies of the Castelnuovo-Mumford regularity of squarefree monomial ideals. Our focus is on bounds and exact values for the regularity in terms of combinatorial data from associated simplicial complexes and/or hypergraphs.Dedicated to Tony Geramita, a great teacher, colleague and friend. IntroductionCastelnuovo-Mumford regularity (or simply regularity) is an important invariant in commutative algebra and algebraic geometry that governs the computational complexity of ideals, modules and sheaves. Computing or finding bounds for the regularity is a difficult problem. Many simply stated questions and conjectures have been verified only in very special cases. For instance, the Eisenbud-Goto conjecture, which states that the regularity of a projective variety is bounded by the difference between its degree and codimension, has been proved only for arithmetic Cohen-Macaulay varieties, curves and surfaces.The class of squarefree monomial ideals is a classical object in commutative algebra, with strong connections to topology and combinatorics, which continues to inspire much of current research. During the last few decades, advances in computer technology and speed of computation have drawn many researchers' attention toward problems and questions involving this class of ideals. Investigating the regularity of squarefree monomial ideals has evolved to be a highly active research topic in combinatorial commutative algebra.In this paper, we survey a number of recent studies on the regularity of squarefree monomial ideals. Even for this class of ideals, the list of results on regularity is too large to exhaust. Our focus will be on studies that find bounds and/or compute the regularity of squarefree monomial ideals in terms of combinatorial data from associated simplicial complexes and hypergraphs. Our aim is to provide readers with an adequate overall picture of the problems, results and techniques, and to demonstrate similarities and differences between these studies. At the same time, we hope to motivate interested researchers, and especially graduate students, to start working in this area. We shall showcase results in a timeline to exhibit (if possible) trends in developing new and/or more general results from previous ones.The paper is structured as follows. In the next section we collect basic notation and terminology from commutative algebra and combinatorics that will be used in the paper. This section provides readers who are new to the research area the necessary background to start. More advanced readers can skip this section and go directly to Section 3. In Section 1 arXiv:1310.7912v2 [math.AC]

show abstract

“…Herzog and Hibi [17,Theorem 3.4] first gave a characterization of Cohen-Macaulay bipartite graphs. This result was extended to very well-covered graphs by Crupi, Rinaldo and Terai [6] (see also Constantinescu and Varbaro [5] and Mahmoudi et al [26]). We will give a short proof of their results.…”

Section: Depth and Regularity Of Edge Idealsmentioning

confidence: 76%