4-Path Vertex Cover: Complexity & Algorithms

• 4-path vertex cover is a graph invariant defined as the smallest vertex set whose removal eliminates all paths of length 4, ensuring the remaining graph is P4-free.
• Exact and parameterized algorithms, with running times like O*(1.87042^t), address its NP-completeness and highlight practical benchmarks for solution efficiency.
• Advanced approximation and kernelization techniques, along with reoptimization strategies, offer actionable insights for network security and structural optimization in graphs.

A 4-path vertex cover number is a graph invariant arising from the general $k$-path vertex cover problem, which asks for a minimum-cardinality vertex subset $F \subseteq V$ of a simple graph $G = (V, E)$ such that the induced subgraph $G[V \setminus F]$ contains no path of order 4. The minimum possible cardinality of $F$ (over all feasible covers) is denoted $U_4(G)$ or $\psi_4(G)$ in the literature. This concept generalizes classical vertex cover (the case $k=2$) and connects to hitting set formulations, path decomposition, network security, and structural optimization in graphs.

1. Formal Definition and Properties

Given a simple undirected graph $G = (V, E)$, a 4-path vertex cover (P4VC) is a subset $F \subseteq V$ such that every simple path $P_4$ on four vertices in $G$ contains at least one vertex from $F$. Formally, $$U_4(G) = \min \{ |F| : F \text{ is a 4-path vertex cover of } G \}$$. Removing $F$ from $G$ leaves a subgraph that is $P_4$-free; that is, all connected components are of size at most 3, or have structures (such as isolated vertices, stars, triangles, claws) incompatible with a path of length 4.

$U_4(G)$ is thus the solution to a specific instance of the $k$-hitting set problem, with universe the set of all $P_4$'s in $G$, and subsets the vertex memberships of each path. Known reductions and the combinatorial structure of paths yield both hardness and insight into algorithmic design.

2. Complexity, Hardness, and Exact Algorithms

The minimum 4-path vertex cover problem (MinVCP$_4$) is NP-complete for general graphs and remains so when restricted to cubic planar graphs, bipartite cubic graphs, and $K_{1,4}$-free graphs; it is APX-complete in these settings (Tu, 2022). This inherent computational difficulty motivates parameterized and exact approaches.

Exact algorithms employ reductions to the 4-hitting set problem and branching/iterative compression schemes. For MinVCP$_4$, the best known running times are $O^*(1.87042^t)$ (unweighted) and $O^*(1.97^t)$ (weighted), where $t$ is the size of the cover and $O^*(\cdot)$ suppresses polynomial factors (Tu, 2022). These algorithms capitalize on the relatively local structure of $P_4$'s and advanced branching strategies on the path incidence structure. The parameterized algorithm from (Tsur, 2018) achieves $O^*(2.619^k)$ time for decision variants parameterized by the cover size $k$.

Kernelization techniques produce polynomial kernels: for MinVCP$_4$, a kernel with at most $176t^2 + 166t$ edges is achievable, supporting efficient preprocessing for FPT algorithms (Tu, 2022).

3. Approximation Algorithms and Regular Graphs

Generic $k$-path vertex cover admits a trivial $k$-approximation. For $k = 4$, the best known general approximation is a factor-3 primal-dual algorithm (Tu, 2022). However, substantially improved results exist for $d$-regular graphs:

• For $d$-regular graphs, defective coloring yields an approximation ratio of $$\frac{\lfloor d/2\rfloor (2d-2)}{(\lfloor d/2\rfloor + 1)(d-2)}$$ for $k = 4$ (Zhang et al., 2018).
• For cubic ($d=3$) and $4$-regular graphs, tailored algorithms exploit the uniform degree structure and triangle presence:
  • Cubic graphs: Approx1 achieves a $1.875$-approximation using a combination of defective $(2,1)$-coloring and auxiliary triangle graphs, leveraging maximum independent set approximations for bounded-degree graphs.
  • $4$-regular graphs: Approx2 employs a layered peeling and pairing technique, reaching a $1.875$ ratio in base analysis; refined fractional charging reduces this to $1.852$ (Zhang et al., 2018).

Comparative results are summarized in the following table:

Graph Class	Best Approximation Ratio for P4VC	Technique
General Graphs	3	Primal-dual
$d$-Regular (any)	$$\frac{\lfloor d/2\rfloor (2d-2)}{(\lfloor d/2\rfloor + 1)(d-2)}$$	Defective coloring
Cubic (3-regular)	1.875	Coloring + triangle MIS
4-regular	1.852	Layered peeling

These advances exploit detailed structural decompositions of $G[V \setminus F^*]$ into basic components (triangles, claws, stars), enabling tight ratio analyses and refined bounds via fractional charging.

4. Reoptimization and Dynamic Graphs

For real-world applications where the input graph evolves by small modifications (e.g., vertex insertion modeling network expansion), reoptimization strategies offer accelerated solutions:

• For unweighted graphs, a PTAS is achievable under constant-size vertex insertions by locally enumerating subsets covering new paths and combining with prior optima (Kumar et al., 2019).
• For weighted P4VC on bounded-degree graphs, reoptimization with a precomputed $\rho$-approximation can be improved to $(2-\tfrac{1}{\rho})$-approximation: with $\rho=3$ (the general 3-approximation), one obtains a $5/3$-approximation for P4VC (Kumar et al., 2019).

This approach restricts computation to the BFS region near modifications, yielding polynomial-time correction even in sparse graphs. It highlights the impact of locality and incremental algorithms for practical, frequently updated systems.

5. Connection to Path Cover, Packing, and Related Problems

While the P4VC asks to "hit" all $P_4$'s with a minimal vertex subset, recent work on the complementary maximum packing problem, MPCv$^{(4+)}$ (packing vertex-disjoint long paths), is relevant to understanding covering behavior and lower bounds (Gong et al., 2023). The latest algorithm for MPCv$^{(4+)}$ leverages maximum matching and path-cycle covers in an auxiliary graph, achieving an approximation ratio of $1.874$ in time $O(\min\{m^2 n^2, n^5\})$.

This matching-based framework suggests plausible implications for improved covering algorithms, via structural decomposition of components and augmentation strategies. Both covering and packing formulations illuminate dual aspects of constraint satisfaction in graphs with respect to path structures.

6. Algorithms for Special Graph Classes and Reconfiguration

On trees and special subclasses—such as caterpillars—reconfiguration problems (transforming one P4VC into another via "token sliding") are of notable interest (Hoang, 2022). For trees formed by a "spine" with attached leaves (caterpillars), a polynomial-time algorithm exists for token sliding reconfiguration of P4VCs:

• The algorithm detects "rigid tokens" (vertices in every cover, impossible to move) using region-based characterization in $O(n^3)$ time.
• After removal of rigid tokens, it partitions the graph, and uses matching and ordered sliding to construct a move sequence between covers.
• Complexity is polynomial, contrasting with PSPACE-completeness for general graphs. The approach enables insight into the structure and feasibility of reconfiguration, with potential for extension to broader classes.

7. Applications and Methodological Insights

Applications of P4VC and $U_4(G)$ arise in secure communication design (WSNs), traffic surveillance, and network monitoring, where coverage over paths of specified length models security or resource constraints (Tu, 2022). By covering all 4-node paths, one ensures guaranteed interception or monitoring in communication or transportation settings.

Methodologically, advances integrate:

• Reductions to hitting set or independence/packing problems.
• Defective coloring for fast partition-based heuristics.
• Iterative compression, branching, and local ratio methods for exact and approximation algorithms.
• Kernelization for parameterized optimization.
• Region-based rigidity analysis for reconfiguration.
• Matching-based augmentation for packing and alternative covering algorithms.

Collectively, these approaches advance both theoretical understanding and practical solution capability for the 4-path vertex cover number and its variants.