Source-linked AI summary
Minimum k-path vertex cover
Boštjan Brešar, František Kardoš, Ján Katrenič, Gabriel Semanišin
TL;DR
The paper studies the minimum k-path vertex cover problem, extending vertex cover and motivated in part by wireless sensor network security. It proves NP-completeness, gives tractability and exact results for important graph classes, develops bounds and estimates, and establishes a general bound for ψ_3(G).
Problem
The paper asks how to characterize and compute the minimum vertex set intersecting every path of order k, a generalization motivated partly by wireless sensor network security.
Method
The paper uses reductions from minimum vertex cover, structural analysis of graph classes, and algorithmic and bounding arguments for trees and general graphs.
Results
The problem is NP-complete for every fixed k≥2; trees admit a linear-time optimal algorithm, and ψ_3(G) ≤ (2n + m)/6 for every graph with n vertices and m edges.
Takeaways & Limitations
k-path vertex cover extends vertex cover while remaining tractable on bounded-treewidth graphs such as trees and supporting general upper-bound estimates.
Abstract
from arXiv · showhide
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 \geq 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) \leq (2n + m)/6, for every graph G with n vertices and m edges.
1. Introduction and motivation
The paper introduces k-path vertex cover as a generalization of vertex cover, motivated partly by security requirements in wireless sensor networks. It establishes complexity results, tractability on bounded-treewidth graphs, and upper bounds including a bound for ψ_3(G).
- Definitions and motivation: Wireless sensor network security motivates covers that ensure at least one uncaptured node on each path of length k−1.The k-generalized Canvas scheme combines cryptographic primitives with network topology and distinguishes protected from unprotected devices.
- Definitions and motivation: A k-path vertex cover intersects every path of order k, and ψ_k(G) is the minimum size of such a cover.For k=2, this is the usual minimum vertex cover; ψ_3(G) corresponds to the dissociation number.
- Complexity and tractability: NP-hardness follows for determining ψ_k, because the minimum vertex cover problem reduces to the minimum k-path vertex cover problem.The paper develops this reduction in Section 2.
- Complexity and tractability: Linear-time algorithms apply on graphs with bounded treewidth, including trees, series-parallel graphs, and outerplanar graphs, by Courcelle’s theorem.The paper determines exact values for trees and presents a linear-time optimal algorithm, while also giving a tight upper bound for ψ_3(G) on outerplanar graphs.
- Bounds: ψ_3(G) ≤ (2n + m)/6 for every graph G with n vertices and m edges.The paper also provides several estimates based on vertex degrees.
2. NP-completeness
The paper proves NP-completeness for fixed k≥2 by reducing vertex cover to k-path vertex cover, then derives approximation consequences and a simple k-approximation algorithm.
- Reduction: For k=2, k-path vertex cover coincides with the ordinary vertex cover problem.For k>2, the proof reduces vertex cover to k-path vertex cover.
- NP-completeness: For every fixed integer k≥2, deciding whether a graph has a k-path vertex cover of size at most t is NP-complete.Membership in NP follows because all paths of length k−1 can be checked in polynomial time.
- Reduction: The reduction attaches a path of floor((k−1)/2) new vertices to every original vertex and preserves the minimum cover size.Examples include attaching a leaf for k=3 or 4 and a two-vertex path for k=5 or 6.
- Reduction: Any new vertex selected in a cover can be replaced by its nearest original vertex without losing coverage.This transformation yields a cover consisting only of original vertices, after which uncovered edges would create an unsecured k-vertex path.
- Approximation: It is NP-hard to approximate k-PVCP within 1.3606 for k>2 unless P=NP.The reduction transfers any polynomial-time r-approximation for k-path vertex cover to minimum vertex cover.
- Approximation: A polynomial-time k-approximation repeatedly selects all k vertices of a found k-vertex path and removes them.At least one selected vertex belongs to an optimal solution; for fixed k, path finding uses a known randomized algorithm with running time O(2^k n^O(1)).
3. Path vertex cover for trees
For trees, PVCPTree constructs an optimal k-path vertex cover by repeatedly selecting and removing properly rooted subtrees. The algorithm runs in linear time and yields the paper’s sharp tree bound.
- Algorithm: PVCPTree repeatedly adds the root v of a properly rooted subtree T_v to the cover and removes T_v from the tree.A properly rooted subtree contains a k-vertex path, while deleting its root leaves none.
- Correctness: Theorem 2 states that PVCPTree returns an optimal k-path vertex cover for every tree.The proof uses induction: every k-path cover must contain a vertex from the selected subtree.
- Size bound: Each iteration adds one cover vertex while removing a subtree with at least k vertices.This counting argument gives the cardinality bound used for trees.
- Complexity and bound: The algorithm can be implemented in linear time and returns a solution of size at most |V(T)|/k.The paper describes this as a sharp upper bound for arbitrary trees.
4. Outerplanar graphs
The paper derives an upper bound for 3-path vertex covers in outerplanar graphs and shows its sharpness through a construction with large required covers. The proof uses maximal outerplanar supergraphs, Hamiltonian orderings, and black-white vertex structure.
- Upper bound: Theorem 3 establishes an upper bound on ψ3(G) for every outerplanar graph G of order n.The proof reduces to a maximal outerplanar supergraph on the same vertices.
- Upper-bound proof: A maximal outerplanar supergraph is 2-connected, has triangular inner faces, and is Hamiltonian along its outer-face boundary.These properties enable the coloring and cyclic-order argument.
- Sharpness: The paper states that the factor in Theorem 3 is best possible for outerplanar graphs.This conclusion follows from the lower-bound construction.
- Sharpness: The construction in the sharpness argument produces a 2-connected outerplanar graph H on 2n vertices with ψ3(H) ≥ n.The vertices are divided into n pairs, and the pair analysis forces the lower bound.
5. Upper bounds on degree of vertices
The paper develops degree-based upper bounds for minimum k-path vertex covers, including a Caro–Wei-type generalization and constructive bounds for 3-path vertex covers. It also gives a tight bound based on graph order and size.
- A generalized Caro–Wei theorem provides an upper bound for ψ_k(G) for every graph G.
- A random-ordering construction builds a set S by adding each vertex unless at least two neighbors are already present.The probability that vertex v_i enters S is described as 2/(1+d(v_i)).
- The resulting set S induces a 1-degenerated graph, and the construction is combined with a minimum k-path vertex cover of S.
- A linear-time algorithm returns a 3-path vertex cover for arbitrary graphs, with bounds including |V(G)|/2 for maximum degree at most 3.The construction uses repeated protection of high-degree vertices followed by a subcubic-graph procedure.
- The bound based on n and m is tight for suitable graphs when b ≤ a ≤ 2b.The constructions use disjoint unions of 4-cycles and graphs obtained from K6 by removing a perfect matching.