Vertex decomposable graphs and obstructions to shellability

Inspired by several recent papers on the edge ideal of a graph G, we study the equivalent notion of the independence complex of G. Using the tool of vertex decomposability from geometric combinatorics, we show that 5-chordal graphs with no chordless 4-cycles are shellable and sequentially Cohen-Macaulay. We use this result to characterize the obstructions to shellability in flag complexes, extending work of Billera, Myers, and Wachs. We also show how vertex decomposability may be used to show that certain graph constructions preserve shellability.Comment: 13 pages, 3 figures. v2: Improved exposition, added Section 5.2 and additional references. v3: minor corrections for publicatio

Edge-Stable Equimatchable Graphs

A graph $G$ is \emph{equimatchable} if every maximal matching of $G$ has the same cardinality. We are interested in equimatchable graphs such that the removal of any edge from the graph preserves the equimatchability. We call an equimatchable graph $G$ \emph{edge-stable} if $G\setminus {e}$, that is the graph obtained by the removal of edge $e$ from $G$, is also equimatchable for any $e \in E(G)$. After noticing that edge-stable equimatchable graphs are either 2-connected factor-critical or bipartite, we characterize edge-stable equimatchable graphs. This characterization yields an $O(\min(n^{3.376}, n^{1.5}m))$ time recognition algorithm. Lastly, we introduce and shortly discuss the related notions of edge-critical, vertex-stable and vertex-critical equimatchable graphs. In particular, we emphasize the links between our work and the well-studied notion of shedding vertices, and point out some open questions

Chordal and sequentially Cohen-Macaulay clutters

We extend the definition of chordal from graphs to clutters. The resulting family generalizes both chordal graphs and matroids, and obeys many of the same algebraic and geometric properties. Specifically, the independence complex of a chordal clutter is shellable, hence sequentially Cohen-Macaulay; and the circuit ideal of a certain complement to such a clutter has a linear resolution. Minimal non-chordal clutters are also closely related to obstructions to shellability, and we give some general families of such obstructions, together with a classification by computation of all obstructions to shellability on 6 vertices.Comment: 20 pages. v2 fixes typos and improves exposition. v3 attributes prior work on shedding faces by Jonsson. v4 has minor updates for publicatio

Algebraic Properties of Clique Complexes of Line Graphs

Let $H$ be a simple undirected graph and $G=\mathrm{L}(H)$ be its line graph. Assume that $\Delta(G)$ denotes the clique complex of $G$. We show that $\Delta(G)$ is sequentially Cohen-Macaulay if and only if it is shellable if and only if it is vertex decomposable. Moreover if $\Delta(G)$ is pure, we prove that these conditions are also equivalent to being strongly connected. Furthermore, we state a complete characterizations of those $H$ for which $\Delta(G)$ is Cohen-Macaulay, sequentially Cohen-Macaulay or Gorenstein. We use these characterizations to present linear time algorithms which take a graph $G$, check whether $G$ is a line graph and if yes, decide if $\Delta(G)$ is Cohen-Macaulay or sequentially Cohen-Macaulay or Gorenstein

On graphs admitting two disjoint maximum independent sets

An independent set A is maximal if it is not a proper subset of an independent set, while A is maximum if it has a maximum size. The problem of whether a graph has a pair of disjoint maximal independent sets was introduced by C. Berge in early 70's. The class of graphs for which every induced subgraph admits two disjoint maximal independent sets was characterized in (Shaudt, 2015). It is known that deciding whether a graph has two disjoint maximal independent sets is a NP-complete problem (Henning et al., 2009). In this paper, we are focused on finding conditions ensuring the existence of two disjoint maximum independent sets.Comment: 12 pages, 4 figure

Cohen-Macaulay oriented graphs with large girth

We classify the Cohen-Macaulay weighted oriented graphs whose underlying graphs have girth at least $5$.Comment: We correct typos in Lemma 2.

Dominating induced matchings of finite graphs and regularity of edge ideals

The regularity of an edge ideal of a finite simple graph $G$ is at least the induced matching number of $G$ and is at most the minimum matching number of $G$. If $G$ possesses a dominating inuduced matching, i.e., an induced matching which forms a maximal matching, then the induced matching number of $G$ is equal to the minimum matching number of $G$. In the present paper, from viewpoints of both combinatorics and commutative algebra, finite simple graphs with dominating induced matchings will be mainly studied.Comment: 23 pages, v2:minor changes, to appear in Journal of Algebraic Combinatoric

A Topological proof that O2O_2 is 22-MCFL

We give a new proof of Salvati's theorem that the group language $O_2$ is $2$ multiple context free. Unlike Salvati's proof, our arguments do not use any idea specific to two-dimensions. This raises the possibility that the argument might generalize to $O_n$.Comment: 11 figures, 2 table

Obstructions to convexity in neural codes

How does the brain encode spatial structure? One way is through hippocampal neurons called place cells, which become associated to convex regions of space known as their receptive fields: each place cell fires at a high rate precisely when the animal is in the receptive field. The firing patterns of multiple place cells form what is known as a convex neural code. How can we tell when a neural code is convex? To address this question, Giusti and Itskov identified a local obstruction, defined via the topology of a code's simplicial complex, and proved that convex neural codes have no local obstructions. Curto et al. proved the converse for all neural codes on at most four neurons. Via a counterexample on five neurons, we show that this converse is false in general. Additionally, we classify all codes on five neurons with no local obstructions. This classification is enabled by our enumeration of connected simplicial complexes on 5 vertices up to isomorphism. Finally, we examine how local obstructions are related to maximal codewords (maximal sets of neurons that co-fire). Curto et al. proved that a code has no local obstructions if and only if it contains certain "mandatory" intersections of maximal codewords. We give a new criterion for an intersection of maximal codewords to be non-mandatory, and prove that it classifies all such non-mandatory codewords for codes on up to 5 neurons.Comment: 21 pages, 1 table; published versio

A construction of sequentially Cohen-Macaulay graphs

For every simple graph $G$, a class of multiple clique cluster-whiskered graphs $G^{md}$ is introduced, and it is shown that all graphs $G^{md}$ are vertex decomposable, thus the independence simplicial complex ${\rm Ind}\,G^{md}$ is sequentially Cohen-Macaulay; the properties of the graphs $G^{md}$ and the clique-whiskered graph $G^\pi$ are studied, including the enumeration of facets of the complex ${\rm Ind}\, G^{\pi}$ and, the calculation of Betti numbers of the cover ideal $I_c(G^{md})$.Comment: 14 pages; the main construction and the main rsults are improved; the part on strong shellable property is removed fromthis pape