Minimum vertex cover in ball graphs through local search

Using local search method, this paper provides a polynomial time approximation scheme for the minimum vertex cover problem on \(d\) -dimensional ball graphs where \(d \ge 3\) . The key to the proof is a new separator theorem for ball graphs in higher dimensional space.     

Minimum vertex cover problem for coupled interdependent networks with cascading failures

This paper defines and analyzes a generalization of the classical minimum vertex cover problem to the case of two-layer interdependent networks with cascading node failures that can be caused by two common types of interdependence. Previous studies on interdependent networks mainly addressed the issues of cascading failures from a numerical simulations perspective, whereas this paper proposes an exact optimization-based approach for identifying a minimum-cardinality set of nodes, whose deletion would effectively disable both network layers through cascading failure mechanisms. We analyze the computational complexity and linear 0–1 formulations of the defined problems, as well as prove an LP approximation ratio result that generalizes the well-known 2-approximation for the classical minimum vertex cover problem. In addition, we introduce the concept of a “depth of cascade” (i.e., the maximum possible length of a sequence of cascading failures for a given interdependent network) and show that for any problem instance this parameter can be explicitly derived via a polynomial-time procedure.     

Minimum augmentation of local edge-connectivity between vertices and vertex subsets in undirected graphs

Given an undirected multigraph G=(V,E), a family W of sets W⊆V of vertices (areas), and a requirement function r:W→Z⁺ (where Z⁺ is the set of nonnegative integers), we consider the problem of augmenting G by the smallest number of new edges so that the resulting graph has at least r(W) edge-disjoint paths between v and W for every pair of a vertex v∈V and an area W∈W. So far this problem was shown to be NP-hard in the uniform case of r(W)=1 for each W∈W, and polynomially solvable in the uniform case of r(W)=r?2 for each W∈W. In this paper, we show that the problem can be solved in time, even if r(W)?2 holds for each W∈W, where n=|V|, m=|{{u,v}|(u,v)∈E}|, p=|W|, and r^*=max{r(W)∣W∈W}.     

PTAS for the minimum k-path connected vertex cover problem in unit disk graphs

In the Minimum k-Path Connected Vertex Cover Problem (MkPCVCP), we are given a connected graph G and an integer k ≥ 2, and are required to find a subset C of vertices with minimum cardinality such that each path with length k ? 1 has a vertex in C, and moreover, the induced subgraph G[C] is connected. MkPCVCP is a generalization of the minimum connected vertex cover problem and has applications in many areas such as security communications in wireless sensor networks. MkPCVCP is proved to be NP-complete. In this paper, we give the first polynomial time approximation scheme (PTAS) for MkPCVCP in unit disk graphs, for every fixed k ≥ 2.     

Minor‐order obstructions for the graphs of vertex cover 6

We provide for the first time, a complete list of forbidden minors (obstructions) for the family of graphs with vertex cover 6. This study shows how to limit both the search space of graphs and improve the efficiency of an obstruction checking algorithm when restricted to k–VERTEX COVER graph families. In particular, our upper bounds 2k + 1 (2k + 2) on the maximum number of vertices for connected (disconnected) obstructions are shown to be sharp for all k > 0. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 163–178, 2002     

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.     

Semi-hyper-connected vertex transitive graphs

A graph G is said to be semi-hyper-connected if the removal of every minimum cut of G creates exactly two connected components. In this paper, we characterize semi-hyper-connected vertex transitive graphs, in particular Cayley graphs.     

Nonrepetitive vertex colorings of graphs

We prove new upper bounds on the Thue chromatic number of an arbitrary graph and on the facial Thue chromatic number of a plane graph in terms of its maximum degree.     

Minimum cycle cover and Chinese postman problems on mixed graphs with bounded tree-width

Let M=(V,E,A) be a mixed graph with vertex set V, edge set E and arc set A. A cycle cover of M is a family C={C₁,…,C_k} of cycles of M such that each edge/arc of M belongs to at least one cycle in C. The weight of C is . The minimum cycle cover problem is the following: given a strongly connected mixed graph M without bridges, find a cycle cover of M with weight as small as possible. The Chinese postman problem is: given a strongly connected mixed graph M, find a minimum length closed walk using all edges and arcs of M. These problems are NP-hard. We show that they can be solved in polynomial time if M has bounded tree-width.     

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).     

Minimum broadcast graphs

Let $\text{K}$ be a class of finite algebras closed under subalgebras, homomorphic images and finite direct products. It is shown that $\text{K}$ is isomorphic to the class of bounded distributive lattices if and only if $\text{K}$ is generated by a lattice-primal algebra.     

Minimum K-hamiltonian graphs

We consider two new classes of graphs arising from reliability considerations in network design. We want to construct graphs with a minimum number of edges which remain Hamiltonian after k edges (or k vertices) have been removed. A simple construction is presented for the case when k is even. We show that it is minimum k-edge Hamiltonian. On the other hand, Chartrand and Kapoor previously proved that this class of graphs was also minimum k-vertex Hamiltonian. The case when k is large (odd or even) is also considered. Some results about directed graphs are also presented.