Standard graded vertex cover algebras, cycles and leaves

The aim of this paper is to characterize simplicial complexes which have standard graded vertex cover algebras. This property has several nice consequences for the squarefree monomial ideals defining these algebras. It turns out that such simplicial complexes are closely related to a range of hypergraphs which generalize bipartite graphs and trees. These relationships allow us to obtain very general results on standard graded vertex cover algebras which cover previous major results on Rees algebras of squarefree monomial ideals.

     

Symbolic powers of monomial ideals and vertex cover algebras

We introduce and study vertex cover algebras of weighted simplicial complexes. These algebras are special classes of symbolic Rees algebras. We show that symbolic Rees algebras of monomial ideals are finitely generated and that such an algebra is normal and Cohen-Macaulay if the monomial ideal is squarefree. For a simple graph, the vertex cover algebra is generated by elements of degree 2, and it is standard graded if and only if the graph is bipartite. We also give a general upper bound for the maximal degree of the generators of vertex cover algebras.     

Unmixed bipartite graphs and sublattices of the Boolean lattices

The correspondence between unmixed bipartite graphs and sublattices of the Boolean lattice is discussed. By using this correspondence, we show existence of squarefree quadratic initial ideals of toric ideals arising from minimal vertex covers of unmixed bipartite graphs.     

笛卡尔乘积图里的最小k-路点覆盖

对于子集$S\subseteq V(G)$,如果图$G$里的每一条$k$路都至少包含$S$中的一个点,那么我们称集合$S$是图$G$的一个$k$-路点覆盖.很明显,这个子集并不唯一.我们称最小的$k$-路点覆盖的基数为$k$-路点覆盖数, 记作$\psi_k(G)$.本文给出了一些笛卡尔乘积图上$\psi_k(G)$值的上界或下界.     

An Ant Colony Optimization Algorithm for the Minimum Weight Vertex Cover Problem

Given an undirected graph and a weighting function defined on the vertex set, the minimum weight vertex cover problem is to find a vertex subset whose total weight is minimum subject to the premise that the selected vertices cover all edges in the graph. In this paper, we introduce a meta-heuristic based upon the Ant Colony Optimization (ACO) approach, to find approximate solutions to the minimum weight vertex cover problem. In the literature, the ACO approach has been successfully applied to several well-known combinatorial optimization problems whose solutions might be in the form of paths on the associated graphs. A solution to the minimum weight vertex cover problem however needs not to constitute a path. The ACO algorithm proposed in this paper incorporates several new features so as to select vertices out of the vertex set whereas the total weight can be minimized as much as possible. Computational experiments are designed and conducted to study the performance of our proposed approach. Numerical results evince that the ACO algorithm demonstrates significant effectiveness and robustness in solving the minimum weight vertex cover problem.     

On complete intersection toric ideals of graphs

We study the graphs G for which their toric ideals I _G are complete intersections. In particular, we prove that for a connected graph G such that I _G is a complete intersection all of its blocks are bipartite except for at most two. We prove that toric ideals of graphs which are complete intersections are circuit ideals. In this case, the generators of the toric ideal correspond to even cycles of G except of at most one generator, which corresponds to two edge disjoint odd cycles joint at a vertex or with a path. We prove that the blocks of these graphs satisfy the odd cycle condition. Finally, we characterize all complete intersection toric ideals of graphs which are normal.     

Regularity and projective dimension of edge ideals

In this paper, we study the regularity and projective dimension of edge ideals. We provide two upper bounds for the regularity of edge ideals of vertex decomposable graphs in terms of the induced matching number and the number of cycles. Also, we generalize one of the upper bounds given by Dao and Schweig for the projective dimension of hypergraphs.     

The vertex (edge) independence number, vertex (edge) cover number and the least eigenvalue of a graph

In this paper we characterize the unique graph whose least eigenvalue attains the minimum among all graphs of a fixed order and a given vertex (edge) independence number or vertex (edge) cover number, and get some bounds for the vertex (edge) independence number, vertex (edge) cover number of a graph in terms of the least eigenvalue of the graph.     

Complexity and approximation results for the connected vertex cover problem in graphs and hypergraphs

We study a variation of the vertex cover problem where it is required that the graph induced by the vertex cover is connected. We prove that this problem is polynomial in chordal graphs, has a PTAS in planar graphs, is APX-hard in bipartite graphs and is 5/3-approximable in any class of graphs where the vertex cover problem is polynomial (in particular in bipartite graphs). Finally, dealing with hypergraphs, we study the complexity and the approximability of two natural generalizations.     

The cover pebbling number of graphs

A pebbling move on a graph consists of taking two pebbles off of one vertex and placing one pebble on an adjacent vertex. In the traditional pebbling problem we try to reach a specified vertex of the graph by a sequence of pebbling moves. In this paper we investigate the case when every vertex of the graph must end up with at least one pebble after a series of pebbling moves. The cover pebbling number of a graph is the minimum number of pebbles such that however the pebbles are initially placed on the vertices of the graph we can eventually put a pebble on every vertex simultaneously. We find the cover pebbling numbers of trees and some other graphs. We also consider the more general problem where (possibly different) given numbers of pebbles are required for the vertices.     

给定度序列图的覆盖成本和反向覆盖成本的研究

具有n个顶点且度序列为(m,2,…,2,1,…,1)(1的重数为m)的连通图不止一个(这些图均为树),而每个树对应唯一一个段序列(l1,l2,…,lm).通过对任意一树移动最长段的悬挂点到最短段悬挂点的方式得到另一树,比较前后两树的覆盖成本和反向覆盖成本,给出了具有最小覆盖成本和反向覆盖成本的极树,并且进一步给出了取得...     

Squarefree Vertex Cover Algebras

In this paper we introduce squarefree vertex cover algebras and exhibit a duality for them. We study the question when these algebras are standard graded and when these algebras coincide with the ordinary vertex cover algebras. It is shown that this is the case for simplicial complexes corresponding to principal Borel sets. Moreover, the generators of these algebras are explicitly described.     

The primitive ideals of the Cuntz–Krieger algebra of a row-finite higher-rank graph with no sources

We catalogue the primitive ideals of the Cuntz–Krieger algebra of a row-finite higher-rank graph with no sources. Each maximal tail in the vertex set has an abelian periodicity group of finite rank at most that of the graph; the primitive ideals in the Cuntz–Krieger algebra are indexed by pairs consisting of a maximal tail and a character of its periodicity group. The Cuntz–Krieger algebra is primitive if and only if the whole vertex set is a maximal tail and the graph is aperiodic.     

Threshold and complexity results for the cover pebbling game

Given a configuration of pebbles on the vertices of a graph, a pebbling move is defined by removing two pebbles from some vertex and placing one pebble on an adjacent vertex. The cover pebbling number of a graph, γ(G), is the smallest number of pebbles such that through a sequence of pebbling moves, a pebble can eventually be placed on every vertex simultaneously, no matter how the pebbles are initially distributed. We determine Bose-Einstein and Maxwell-Boltzmann cover pebbling thresholds for the complete graph. Also, we show that the cover pebbling decision problem is NP-complete.     

Representations of centrally-extended Lie algebras over differential operators and vertex algebras

We construct irreducible modules of centrally-extended classical Lie algebras over left ideals of the algebra of differential operators on the circle, through certain irreducible modules of centrally-extended classical Lie algebras of infinite matrices with finite number of nonzero entries. The structures of vertex algebras associated with the vacuum representations of these algebras are determined. Moreover, we prove that under certain conditions, the highest-weight irreducible modules of centrally-extended classical Lie algebras of infinite matrices with finite number of nonzero entries naturally give rise to the irreducible modules of the simple quotients of these vertex algebras. From vertex algebra and its representation point of view, our results with positive integral central charge are high-order differential operator analogues of the well-known WZW models in conformal field theory associated with affine Kac-Moody algebras. Indeed, when the left ideals are the algebra of differential operators, our Lie algebras do contain affine Kac-Moody algebras as subalgebras and our results restricted on them are exactly the representation contents in WZW models. Similar results with negative central charge are also obtained.     

Generalized binomial edge ideals

This paper studies a class of binomial ideals associated to graphs with finite vertex sets. They generalize the binomial edge ideals, and they arise in the study of conditional independence ideals. A Gröbner basis can be computed by studying paths in the graph. Since these Gröbner bases are square-free, generalized binomial edge ideals are radical. To find the primary decomposition a combinatorial problem involving the connected components of subgraphs has to be solved. The irreducible components of the solution variety are all rational.     

Graphs with equal eternal vertex cover and eternal domination numbers

Mobile guards on the vertices of a graph are used to defend it against an infinite sequence of attacks on either its vertices or its edges. If attacks occur at vertices, this is known at the eternal domination problem. If attacks occur at edges, this is known as the eternal vertex cover problem. We focus on the model in which all guards can move to neighboring vertices in response to an attack. Motivated by the question of which graphs have equal eternal vertex cover and eternal domination numbers, a number of results are presented; one of the main results of the paper is that the eternal vertex cover number is greater than the eternal domination number (in the all-guards move model) in all graphs of minimum degree at least two.     

Combinatorial symbolic powers

Symbolic powers are studied in the combinatorial context of monomial ideals. When the ideals are generated by quadratic squarefree monomials, the generators of the symbolic powers are obstructions to vertex covering in the associated graph and its blowups. As a result, perfect graphs play an important role in the theory, dual to the role played by perfect graphs in the theory of secants of monomial ideals. We use Gröbner degenerations as a tool to reduce questions about symbolic powers of arbitrary ideals to the monomial case. Among the applications are a new, unified approach to the Gröbner bases of symbolic powers of determinantal and Pfaffian ideals.     

A note on the last new vertex visited by a random walk

A “cover tour” of a connected graph G from a vertex x is a random walk that begins at x, moves at each step with equal probability to any neighbor of its current vertex, and ends when it has hit every vertex of G. The cycle C_n is well known to have the curious property that a cover tour from any vertex is equally likely to end at any other vertex; the complete graph K_n shares this property, trivially, by symmetry. Ronald L. Graham has asked whether there are any other graphs with this property; we show that there are not. © 1993 John Wiley & Sons, Inc.