K-path Vertex Cover Research Articles

A Survey on the k-Path Vertex Cover Problem

Given an integer k ≥ 2, a k-path is a path on k vertices. A set of vertices in a graph G is called a k-path vertex cover if it includes at least one vertex of every k-path of G. A minimum k-path vertex cover in G is a k-path vertex cover having the smallest possible number of vertices and its cardinality is called the k-path vertex cover number of G. In the k-path vertex cover problem, the goal is to find a minimum k-path vertex cover in a given graph. In this paper, we present a brief survey of the current state of the art in the study of the k-path vertex cover problem and the k-path vertex cover number.

3-path vertex cover and dissociation number of hexagonal graphs

A subset P of vertices of a graph G is called a k-path vertex cover if every path of order k in G contains at least one vertex from P. The cardinality of a minimum k-path vertex cover is called the k-path vertex cover number of G, and is denoted by ?k(G). It is known that the problem of finding a minimum 3-path vertex cover is NP-hard for planar graphs. In this paper we establish an upper bound for ?3(G), where G is from an important family of planar graphs, called hexagonal graphs, arising from real world applications.

A polynomial-time algorithm of finding a minimum k-path vertex cover and a maximum k-path packing in some graphs

For a graph G and a positive integer k, a subset C of vertices of G is called a k-path vertex cover if C intersects all paths of k vertices in G. The cardinality of a minimum k-path vertex cover is denoted by $$\beta _{P_k}(G)$$ . For a graph G and a positive integer k, a subset M of pairwise vertex-disjoint paths of k vertices in G is called a k-path packing. The cardinality of a maximum k-path packing is denoted by $$\mu _{P_k}(G)$$ . In this paper, we describe some graphs, having equal values of $$\beta _{P_k}$$ and $$\mu _{P_k}$$ , for $$k \ge 5$$ , and present polynomial-time algorithms of finding a minimum k-path vertex cover and a maximum k-path packing in such graphs.

Microalbuminuria detection: comparisons of two differently semiquantitative urine strip tests with a quantitative automated chemistry analyzer

A k-path vertex cover (VCPk) is a vertex set C of graph G such that every path of G on k vertices has at least one vertex in C. Because of its background in keeping data integrality of a network, minimum VCPk problem (MinVCPk) has attracted a lot of researches in recent years. This paper studies the minimum weight connected VCPk problem (MinWCVCPk), in which every vertex has a weight and the VCPk found by the algorithm induces a connected subgraph and has the minimum weight. It is known that MinWCVCPk is set-cover-hard. We present two polynomial-time approximation algorithms for MinWCVCP3. The first one is a greedy algorithm achieving approximation ratio 3ln n. The difficulty lies in its analysis dealing with a non-submodular potential function. The second algorithm is a 2-stage one, finding a VCP3 in the first stage and then adding more vertices for connection. We show that its approximation ratio is at most lnδmax+4+ln2, where δmax is the maximum degree of the graph. Considering the inapproximability of this problem, this ratio is asymptotically tight.

Approximation algorithms for minimum weight connected 3-path vertex cover

A k-path vertex cover (VCPk) is a vertex set C of graph G such that every path of G on k vertices has at least one vertex in C. Because of its background in keeping data integrality of a network, minimum VCPk problem (MinVCPk) has attracted a lot of researches in recent years. This paper studies the minimum weight connected VCPk problem (MinWCVCPk), in which every vertex has a weight and the VCPk found by the algorithm induces a connected subgraph and has the minimum weight. It is known that MinWCVCPk is set-cover-hard. We present two polynomial-time approximation algorithms for MinWCVCP3. The first one is a greedy algorithm achieving approximation ratio 3ln n. The difficulty lies in its analysis dealing with a non-submodular potential function. The second algorithm is a 2-stage one, finding a VCP3 in the first stage and then adding more vertices for connection. We show that its approximation ratio is at most lnδmax+4+ln2, where δmax is the maximum degree of the graph. Considering the inapproximability of this problem, this ratio is asymptotically tight.

A simpler PTAS for connected k-path vertex cover in homogeneous wireless sensor network

Because of its application in the field of security in wireless sensor networks, k-path vertex cover ( $$\hbox {VCP}_k$$ ) has received a lot of attention in recent years. Given a graph $$G=(V,E)$$ , a vertex set $$C\subseteq V$$ is a k-path vertex cover ( $$\hbox {VCP}_k$$ ) of G if every path on k vertices has at least one vertex in C, and C is a connected k-path vertex cover of G ( $$\hbox {CVCP}_k$$ ) if furthermore the subgraph of G induced by C is connected. A homogeneous wireless sensor network can be modeled as a unit disk graph. This paper presents a new PTAS for $$\hbox {MinCVCP}_k$$ on unit disk graphs. Compared with previous PTAS given by Liu et al., our method not only simplifies the algorithm and reduces the time-complexity, but also simplifies the analysis by a large amount.

The k-path vertex cover in Cartesian product graphs and complete bipartite graphs

For a graph G and a positive integer k, a subset S of vertices of G is called a k-path vertex cover if S intersects all paths of order k in G. The cardinality of a minimum k-path vertex cover is denoted by ψk(G), and called the k-path vertex cover number of G. In this paper, we study some Cartesian product graphs and give several estimations and the exact values of ψk(G).

PTAS for [formula omitted]-free node deletion problems in disk graphs

For a set H of graphs, a graph G is H-free if G does not contain any subgraph isomorphic to some graph in H. In this paper, we study the minimum H-free node deletion problem (MinHFND) and the maximum H-free node set problem (MaxHFNS), which include a lot of extensively-studied problems such as the minimum k-path vertex cover problem, the dissociation number problem, and the minimum degree bounded node deletion problem. For a large class of H, PTASs are given for MinHFND and MaxHFNS on disk graphs whose heterogeneity is bounded by a constant, where the heterogeneity of a disk graph is the ratio of the maximum radius to the minimum radius of disks.

PTAS for minimum k-path vertex cover in ball graph

A vertex set F is a k-path vertex cover (VCPk) of graph G if every path of G on k vertices contains at least one vertex from F. A graph G is a d-dimensional ball graph if each vertex of G corresponds to a ball in Rd, two vertices are adjacent in G if and only if their corresponding balls have nonempty intersection. The heterogeneity of a ball graph is defined to be rmax/rmin, where rmax and rmin are the largest radius and the smallest radius of those balls, respectively. In this paper, we present a PTAS for the minimum VCPk problem in a ball graph whose heterogeneity is bounded by a constant.

Approximation algorithms for minimum (weight) connected [formula omitted]-path vertex cover

A vertex subset C of a connected graph G is called a connected k-path vertex cover (CVCPk) if every path on k vertices contains at least one vertex from C, and the subgraph of G induced by C is connected. This concept originated in the field of security and supervisory control. This paper studies the minimum (weight) CVCPk problem. We first show that the minimum weight CVCPk problem can be solved in time O(n) when the graph is a tree, and can be solved in time O(rn) when the graph is a uni-cyclic graph whose unique cycle has length r, where n is the number of vertices. Making use of the algorithm on trees, we present a k-approximation algorithm for the minimum (cardinality) CVCPk problem under the assumption that the graph has girth at least k. An example is given showing that performance ratio k is asymptotically tight for our algorithm.

A faster FPT algorithm for 3-path vertex cover

The k-path vertex cover of a graph G is a subset S of vertices of G such that every path on k vertices in G contains at least one vertex from S. Denote by ψk(G) the minimum cardinality of a k-path vertex cover set in G. The minimum k-path vertex cover problem (k-PVCP) is to find a k-path vertex cover of size ψk(G). In this paper we present an FPT algorithm to the 3-PVCP with runtime O(1.8172snO(1)) on a graph with n vertices. The algorithm constructs a 3-path vertex cover of size at most s in a given graph G, or reports that no such 3-path vertex cover exists in G. This improves previous O(2snO(1)) upper bound by Tu [5] and O(1.882snO(1)) upper bound by Wu [13].

The [formula omitted]-path vertex cover of rooted product graphs

A subset S of vertices of a graph G is called a k-path vertex cover if every path of order k in G contains at least one vertex from S. Denote by ψk(G) the minimum cardinality of a k-path vertex cover in G. In this article a lower and an upper bound for ψk of the rooted product graphs are presented. Two characterizations are given when those bounds are attained. Moreover ψ2 and ψ3 are exactly determined. As a consequence the independence and the dissociation number of the rooted product are given.

On the weighted [formula omitted]-path vertex cover problem

A subset S of vertices of a graph G is called a k-path vertex cover if every path of order k in G contains at least one vertex from S. The cardinality of a minimum k-path vertex cover is called the k-path vertex cover number of a graph G, denoted by ψk(G). It is known that the minimum k-path vertex cover problem (k-PVCP) is NP-hard for all k≥2. In this paper we consider the weighted version of a k-PVCP (k-WPVCP), in which vertices are given weights, and the problem is to find a minimum weight set in G such that the graph obtained by deleting this set from G has no Pk. This problem was briefly introduced by Tu and Zhou (2011) but has not been studied to much extent. We give solutions and efficient algorithms for the k-WPVCP for some special classes of graphs. In particular, for complete graphs and cycles, and most importantly an algorithm that computes the k-path vertex cover number of a tree with time complexity O(k⋅|V(G)|) is presented.

On the [formula omitted]-path vertex cover of some graph products

A subset S of vertices of a graph G is called a k-path vertex cover if every path of order k in G contains at least one vertex from S. Denote by ψk(G) the minimum cardinality of a k-path vertex cover in G. In this paper, improved lower and upper bounds for ψk of the Cartesian and the strong product of paths are derived. It is shown that for ψ3 those bounds are tight. For the lexicographic product bounds are presented for ψk, moreover ψ2 and ψ3 are exactly determined for the lexicographic product of two arbitrary graphs. As a consequence the independence and the dissociation number of the lexicographic product are given.