Minimum k-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 \psi_k(G) the minimum cardinality of a k-path vertex cover in G. It is shown that the problem of determining \psi_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 \psi_k(G) and provide several estimations and exact values of \psi_k(G). We also prove that \psi_3(G) \leq (2n + m)/6, for every graph G with n vertices and m edges.

💡 Research Summary

The paper introduces the concept of a k‑path vertex cover in a finite simple graph G. A set S ⊆ V(G) is a k‑path vertex cover if every path of order k (i.e., containing exactly k vertices) includes at least one vertex of S. The minimum cardinality of such a set is denoted ψₖ(G). The authors investigate the computational complexity of determining ψₖ(G), provide exact algorithms for special graph classes, and derive several upper bounds that depend on structural parameters such as the number of vertices, edges, and vertex degrees.

Complexity.
For k = 2 the problem coincides with the classical vertex‑cover problem, which is NP‑complete. The authors extend this hardness to every fixed integer k ≥ 2. The reduction starts from an arbitrary graph G and constructs a new graph G′ by attaching to each original vertex a path of length ⌊(k‑1)/2⌋. They prove that a minimum vertex cover of G corresponds bijectively to a minimum k‑path vertex cover of G′, establishing NP‑completeness of the decision version (k‑PVC). Moreover, because the reduction preserves approximation ratios, any r‑approximation for k‑PVC would yield an r‑approximation for vertex cover, implying that, unless P = NP, no polynomial‑time algorithm can approximate k‑PVC within a factor better than 1 for k > 2.

Trees.
Despite the general hardness, the problem becomes tractable on trees. Using Courcelle’s theorem one could obtain a linear‑time algorithm for bounded‑treewidth graphs, but the authors present a concrete dynamic‑programming procedure called PVCPTree. The algorithm repeatedly finds a properly rooted subtree Tᵥ that contains a k‑vertex path while its parent does not, adds the root v to the cover, and removes Tᵥ from the tree. By induction they show that every optimal solution must contain such a root, so the greedy removal yields an optimal cover. The algorithm runs in O(|V|) time and guarantees ψₖ(T) ≤ |V(T)|/k for any tree T. This bound is tight for paths.

Outerplanar graphs (k = 3).
The authors focus on ψ₃ for outerplanar graphs. For a maximal outerplanar graph H (which is 2‑connected and Hamiltonian), they color vertices of degree 2 white and the rest black. White vertices form an independent set, and black vertices constitute a 3‑path vertex cover. If there are no consecutive black vertices (i.e., all edges incident to the outer face are “good”), exactly half the vertices are black, giving ψ₃(H) ≤ n/2. When a “bad” edge (two consecutive black vertices) exists, they locate a triangular face adjacent to the outer face that contains the bad edge and a chain of alternating white/black vertices. By removing this chain and applying induction on the remaining graph, they again obtain a cover of size at most n/2. The bound is shown to be best possible by constructing outerplanar graphs where ψ₃ reaches n/2.

Degree‑based upper bounds.
Building on the Caro‑Wei bound for ordinary vertex cover (ψ₂ ≤ |V| − ∑1/(1+d(v))), the paper generalizes to arbitrary k. By randomly ordering the vertices and adding a vertex v to the cover unless at most one of its neighbors has already been selected, the probability that v ends up in the cover is 2/(1+d(v)). The expected size of the resulting set S is therefore Σ 2/(1+d(v)). Since S is 1‑degenerate (a forest), Theorem 2 (ψₖ(F) ≤ |V(F)|/k for forests) yields

 ψₖ(G) ≤ |V| − (k‑1)/k · ∑ 1/(1+d(v)).

For k = 3 a refined bound involving pairs of adjacent vertices is also given, improving the estimate when high‑degree vertices are adjacent.

A simple universal bound for ψ₃.
One of the most striking results is the inequality

 ψ₃(G) ≤ (2n + m)/6,

where n = |V(G)| and m = |E(G)|. The proof combines degree counting with the observation that each uncovered 3‑path forces a certain amount of “charge” to be distributed among its vertices and edges. This bound depends only on the basic parameters n and m, making it easy to apply to any graph. It is especially strong for sparse graphs (e.g., trees, where m = n‑1 gives ψ₃ ≤ (3n‑1)/6 ≈ n/2, matching the tree bound).

Implications for wireless sensor networks.
The original motivation stems from secure communication in wireless sensor networks (WSNs). In the k‑generalized Canvás scheme, a protected node must appear on every path of length k‑1 to guarantee data integrity. Minimizing the number of protected nodes is exactly the k‑path vertex cover problem. The linear‑time algorithm for trees is directly applicable to tree‑like sensor topologies, while the outerplanar bound informs planar deployments. The degree‑based bounds give quick estimates for arbitrary network graphs, and the universal ψ₃ ≤ (2n+m)/6 bound can be used as a design guideline when k = 3 (the most common case in practice).

Conclusion.
The paper establishes that determining ψₖ(G) is NP‑complete for all k ≥ 2, provides a linear‑time optimal algorithm for trees, proves a tight ψ₃ ≤ n/2 bound for outerplanar graphs, extends Caro‑Wei type degree bounds to general k, and derives a concise universal inequality ψ₃ ≤ (2n+m)/6. These contributions deepen the theoretical understanding of path‑based vertex covers and offer practical tools for network security design.