Path Vertex Cover Research Articles

A 5k-vertex kernel for 3-path vertex cover

A 3-path vertex cover in a graph is a vertex subset C such that every path of three vertices contains at least one vertex from C. The parameterized 3-path vertex cover problem asks whether a graph has a 3-path vertex cover of size at most k. We give a kernel of 5k vertices for this problem, improving the previously known bounds.

TS-Reconfiguration of $k$-Path Vertex Covers in Caterpillars for $k \geq 4$

A $k$-path vertex cover ($k$-PVC) of a graph $G$ is a vertex subset $I$ such that each path on $k$ vertices in $G$ contains at least one member of $I$. Imagine that a token is placed on each vertex of a $k$-PVC. Given two $k$-PVCs $I, J$ of a graph $G$, the \textsc{$k$-Path Vertex Cover Reconfiguration ($k$-PVCR)} under Token Sliding ($\mathsf{TS}$) problem asks if there is a sequence of $k$-PVCs between $I$ and $J$ where each intermediate member is obtained from its predecessor by sliding a token from some vertex to one of its unoccupied neighbors. This problem is known to be $\mathtt{PSPACE}$-complete even for planar graphs of maximum degree $3$ and bounded treewidth and can be solved in polynomial time for paths and cycles. Its complexity for trees remains unknown. In this paper, as a first step toward answering this question, for $k \geq 4$, we present a polynomial-time algorithm that solves \textsc{$k$-PVCR} under $\mathsf{TS}$ for caterpillars (i.e., trees formed by attaching leaves to a path).

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.

The k‐path vertex cover: General bounds and chordal graphs

AbstractFor an integer , a k‐path vertex cover of a graph is a set that shares a vertex with every path subgraph of order k in G. The minimum cardinality of a k‐path vertex cover is denoted by . We give estimates—mostly upper bounds—on in terms of various parameters, including vertex degrees and the number of vertices and edges. The problem is also considered on chordal graphs and planar graphs.

Approximation algorithm for minimum connected 3-path vertex cover

A vertex subset S of a given graph G=(V,E) is called a connectedk-path vertex cover (CVCPk) if every k-path of G contains at least one vertex from S, and the subgraph of G induced by S is connected. This concept has its background in the field of security and supervisory and the computation of a minimum CVCPk is NP-hard. In this paper, we give a (2α+1∕2)-approximation algorithm for MinCVCP3, where α is the performance ratio of an algorithm for MinVCP3.

Algorithm for Online 3-Path Vertex Cover

A vertex set C of a graph G = (V, E) is a 3-path vertex cover if every path on 3 vertices has at least one vertex in C. This paper studies the online version of the minimum 3-path vertex cover problem, in which vertices are revealed one by one, and one has to determine whether the newly revealed vertex should be chosen into the solution without knowing future information. We show that a natural algorithm has competitive ratio at most Δ, where Δ is the maximum degree of the graph. An example is given showing that the ratio is tight.

Parameterized algorithm for 3-path vertex cover

In the 3-path vertex cover problem, the input is an undirected graph G and an integer k. The goal is to decide whether there is a set of vertices S of size at most k such that every path with 3 vertices in G contains at least one vertex of S. In this paper we give a parameterized algorithm for 3-path vertex cover whose time complexity is O⁎(1.713k). Our algorithm is faster than previous algorithms for this problem.

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.

On a relation between [formula omitted]-path partition and [formula omitted]-path vertex cover

The minimum vertex cover problem and the minimum vertex partition problem are central problems in graph theory and many generalisations are known. Two examples are the minimumk-path vertex cover problem (MkPVCP for short), which asks for a minimum set of vertices covering every path of order k, and the minimumk-path partition problem (MkPPP for short), which asks for a minimum set of paths in a maximal path packing whose every path has at most k vertices.In this paper, we will present a relation between the MkPPP and the MkPVCP. Based on it, we will obtain new bounds for their invariants and a new sufficient condition for NP-hardness of the MkPVCP in terms of forbidden subgraphs.

Exact algorithms for the maximum dissociation set and minimum 3-path vertex cover problems

A dissociation set in a graph G=(V,E) is a vertex subset D such that the subgraph G[D] induced on D has vertex degree at most 1. A 3-path vertex cover in a graph is a vertex subset C such that every path of three vertices contains at least one vertex from C. A vertex set D is a dissociation set if and only if V∖D is a 3-path vertex cover. There are many applications for dissociation sets and 3-path vertex covers. However, it is NP-hard to compute a dissociation set of maximum size or a 3-path vertex cover of minimum size in graphs. Several exact algorithms have been proposed for these two problems and they can be solved in O⁎(1.4658n) time in n-vertex graphs. In this paper, we reveal some interesting structural properties of the two problems, which allow us to solve them in O⁎(1.4656n) time and polynomial space or O⁎(1.3659n) time and exponential space.

Kernelization of the [formula omitted]-path vertex cover problem

The 3-path vertex cover problem is an extension of the well-known vertex cover problem. It asks for a vertex set S⊆V(G) of minimum cardinality such that G−S only contains independent vertices and edges. In this paper we will present a polynomial algorithm which computes two disjoint sets T1,T2 of vertices of G such that (i) for any 3-path vertex cover S′ in G[T2], S′∪T1 is a 3-path vertex cover in G, (ii) there exists a minimum 3-path vertex cover in G which contains T1 and (iii) |T2|≤6⋅ψ3(G[T2]), where ψ3(G) is the cardinality of a minimum 3-path vertex cover and T2 is the kernel of G.

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 [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.

On computing the minimum 3-path vertex cover and dissociation number of graphs

The dissociation number of a graph G is the number of vertices in a maximum size induced subgraph of G with vertex degree at most 1. A k -path vertex cover of a graph G is a subset S of vertices of G such that every path of order k in G contains at least one vertex from S . The minimum 3-path vertex cover is a dual problem to the dissociation number. For this problem, we present an exact algorithm with a running time of O ∗ ( 1.517 1 n ) on a graph with n vertices. We also provide a polynomial time randomized approximation algorithm with an expected approximation ratio of 23 11 for the minimum 3-path vertex cover.

Minimum [formula omitted]-path vertex cover

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 ψ k ( G ) the minimum cardinality of a k -path vertex cover in G . It is shown that the problem of determining ψ k ( G ) is NP-hard for each k ≥ 2 , while for trees the problem can be solved in linear time. We investigate upper bounds on the value of ψ k ( G ) and provide several estimations and exact values of ψ k ( G ) . We also prove that ψ 3 ( G ) ≤ ( 2 n + m ) / 6 , for every graph G with n vertices and m edges.