An upper bound for Cubicity in terms of Boxicity

An axis-parallel b-dimensional box is a Cartesian product R₁×R₂×?×R_b where each R_i (for 1≤i≤b) is a closed interval of the form [a_i,b_i] on the real line. The boxicity of any graph G, is the minimum positive integer b such that G can be represented as the intersection graph of axis-parallel b-dimensional boxes. A b-dimensional cube is a Cartesian product R₁×R₂×?×R_b, where each R_i (for 1≤i≤b) is a closed interval of the form [a_i,a_i+1] on the real line. When the boxes are restricted to be axis-parallel cubes in b-dimension, the minimum dimension b required to represent the graph is called the cubicity of the graph (denoted by ). In this paper we prove that , where n is the number of vertices in the graph. We also show that this upper bound is tight.Some immediate consequences of the above result are listed below:

1.

Planar graphs have cubicity at most 3⌈log₂n⌉.

2.

Outer planar graphs have cubicity at most 2⌈log₂n⌉.

3.

Any graph of treewidth tw has cubicity at most (tw+2)⌈log₂n⌉. Thus, chordal graphs have cubicity at most (ω+1)⌈log₂n⌉ and circular arc graphs have cubicity at most (2ω+1)⌈log₂n⌉, where ω is the clique number.

The above upper bounds are tight, but for small constant factors.     

Properties of vertex cover obstructions

We study properties of the sets of minimal forbidden minors for the families of graphs having a vertex cover of size at most k. We denote this set by O(k-VERTEX COVER) and call it the set of obstructions. Our main result is to give a tight vertex bound of O(k-VERTEX COVER), and then confirm a conjecture made by Liu Xiong that there is a unique connected obstruction with maximum number of vertices for k-VERTEX COVER and this graph is C_2k+1. We also find two iterative methods to generate graphs in O((k+1)-VERTEX COVER) from any graph in O(k-VERTEX COVER).     

Improved approximation of maximum vertex cover

We provide a new LP relaxation of the maximum vertex cover problem and a polynomial-time algorithm that finds a solution within the approximation factor , where is the size of the smallest clique in a given clique-partition of the edge weighting of G.     

A list heuristic for vertex cover

We analyze a list heuristic for the vertex cover problem that handles the vertices in a given static order based on the degree sequence. We prove an approximation ratio of at most for a nonincreasing degree sequence, and show that no ordering can achieve an approximation ratio of less than .     

Toric rings arising from vertex cover ideals

We extend the sortability concept to monomial ideals which are not necessarily generated in one degree and as an application we obtain normal Cohen-Macaulay toric rings attached to vertex cover ideals of graphs. Moreover, we consider a construction on a graph called a clique multi-whiskering which always produces vertex cover ideals with componentwise linear powers.     

An edge-reduction algorithm for the vertex cover problem

An approximation algorithm for the vertex cover problem is proposed with performance ratio on special graphs. On an arbitrary graph, the algorithm guarantees a vertex cover S₁ such that where S^∗ is an optimal cover and ξ is an error bound identified.     

On the powers of vertex cover ideals

Let $S=\mathbb{K}[x_1,\dots ,x_n]$ be a polynomial ring, where $\mathbb{K}$ is a field, and G be a simple graph on n vertices. Let $J(G)?S$ be the vertex cover ideal of G. Herzog, Hibi and Ohsugi have conjectured that all powers of vertex cover ideals of chordal graph are componentwise linear. Here we establish the conjecture for the special case of trees. We also show that if G is a unicyclic vertex decomposable graph, then symbolic powers of $J(G)$ are componentwise linear.     

The cubicity of hypercube graphs

For a graph G, its cubicity is the minimum dimension k such that G is representable as the intersection graph of (axis-parallel) cubes in k-dimensional space. (A k-dimensional cube is a Cartesian product R₁×R₂×?×R_k, where R_i is a closed interval of the form [a_i,a_i+1] on the real line.) Chandran et al. [L.S. Chandran, C. Mannino, G. Oriolo, On the cubicity of certain graphs, Information Processing Letters 94 (2005) 113-118] showed that for a d-dimensional hypercube H_d, . In this paper, we use the probabilistic method to show that . The parameter boxicity generalizes cubicity: the boxicity of a graph G is defined as the minimum dimension k such that G is representable as the intersection graph of axis-parallel boxes in k-dimensional space. Since for any graph G, our result implies that . The problem of determining a non-trivial lower bound for is left open.     

Nearly tight approximation bounds for vertex cover on dense k-uniform k-partite hypergraphs

We establish almost tight upper and lower approximation bounds for the Vertex Cover problem on dense k-uniform k-partite hypergraphs.     

Local and union boxicity

The boxicity $box\left(H\right)$ of a graph $H$ is the smallest integer $d$ such that $H$ is the intersection of $d$ interval graphs, or equivalently, that $H$ is the intersection graph of axis-aligned boxes in ${\mathbb{R}}^d$. These intersection representations can be interpreted as covering representations of the complement $H^c$ of $H$ with co-interval graphs, that is, complements of interval graphs. We follow the recent framework of global, local and folded covering numbers (Knauer and Ueckerdt, 2016) to define two new parameters: the local boxicity ${box}_{?}\left(H\right)$ and the union boxicity $\overline{box}\left(H\right)$ of $H$. The union boxicity of $H$ is the smallest $d$ such that $H^c$ can be covered with $d$ vertex–disjoint unions of co-interval graphs, while the local boxicity of $H$ is the smallest $d$ such that $H^c$ can be covered with co-interval graphs, at most $d$ at every vertex.We show that for every graph $H$ we have ${box}_{?}\left(H\right)\le \overline{box}\left(H\right)\le box\left(H\right)$ and that each of these inequalities can be arbitrarily far apart. Moreover, we show that local and union boxicity are also characterized by intersection representations of appropriate axis-aligned boxes in ${\mathbb{R}}^d$. We demonstrate with a few striking examples, that in a sense, the local boxicity is a better indication for the complexity of a graph, than the classical boxicity.     

On the Cubicity of Interval Graphs

A k-cube (or “a unit cube in k dimensions”) is defined as the Cartesian product where R _i (for 1 ≤ i ≤ k) is an interval of the form [a _i , a _i  + 1] on the real line. The k-cube representation of a graph G is a mapping of the vertices of G to k-cubes such that the k-cubes corresponding to two vertices in G have a non-empty intersection if and only if the vertices are adjacent. The cubicity of a graph G, denoted as cub(G), is defined as the minimum dimension k such that G has a k-cube representation. An interval graph is a graph that can be represented as the intersection of intervals on the real line - i.e., the vertices of an interval graph can be mapped to intervals on the real line such that two vertices are adjacent if and only if their corresponding intervals overlap. We show that for any interval graph G with maximum degree Δ, . This upper bound is shown to be tight up to an additive constant of 4 by demonstrating interval graphs for which cubicity is equal to .     

Cubicity of threshold graphs

We show that the cubicity of a connected threshold graph is equal to ⌈log₂α⌉, where α is its independence number.     

The most vital nodes with respect to independent set and vertex cover

Given an undirected graph with weights on its vertices, the k most vital nodes independent set (k most vital nodes vertex cover) problem consists of determining a set of k vertices whose removal results in the greatest decrease in the maximum weight of independent sets (minimum weight of vertex covers, respectively). We also consider the complementary problems, minimum node blocker independent set (minimum node blocker vertex cover) that consists of removing a subset of vertices of minimum size such that the maximum weight of independent sets (minimum weight of vertex covers, respectively) in the remaining graph is at most a specified value. We show that these problems are NP-hard on bipartite graphs but polynomial-time solvable on unweighted bipartite graphs. Furthermore, these problems are polynomial also on cographs and graphs of bounded treewidth. Results on the non-existence of ptas are presented, too.     

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.     

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.