3-Regular Bipartite Planar Vertex Cover

Updated 13 October 2025

3-regular bipartite planar vertex cover is a combinatorial concept defined on two-colored planar graphs where every vertex has degree three, ensuring a duality between maximum matching and minimum vertex cover.
The study employs a range of methodologies including exact and parameterized algorithms, distributed and combinatorial approximation schemes, as well as algebraic and statistical analyses.
Research reveals that while exact solutions are tractable via LP duality, challenges arise from NP-hard and #P-hard aspects in enumeration and distributed approximation.

A 3-regular bipartite planar vertex cover is a combinatorial object arising in the paper of planar graphs partitionable into two color classes, each vertex having degree three. The problem of finding, analyzing, and counting minimum vertex covers in such graphs is central to areas ranging from distributed computing and commutative algebra to statistical physics and computational complexity. This topic encompasses exact algorithms, parameterized complexity, algebraic invariants, distributed approximation schemes, reconfiguration problems, and direct reductions to #P-hard enumeration tasks.

1. Fundamental Properties and Existence Criteria

A 3-regular bipartite planar graph has two sets of vertices, each of size $n$ , every vertex incident to three edges, and an embedding in the plane without crossings. The existence of such graphs is characterized by degree sequence constraints and planarity criteria.

From (Adams et al., 2018), the necessary and sufficient conditions for the existence of a simple, planar bipartite graph with degree sequences $(3^p \mid 3^p)$ are:

Degree sum equality: %%%%2%%%%.
Euler's inequality: %%%%3%%%%, implying $p \geq 4$ .
Exclusion of exceptional cases: $(3^5 \mid 3^5)$ and $(3^{25} \mid 5^{15})$ cannot be realized as planar graphs despite satisfying the above numerical conditions.

Thus, the minimal size $p=4$ is permitted for 3-regular bipartite planar graphs, while $p=5$ and certain mixed degree sequences are forbidden.

2. Vertex Cover Structure and Duality

König's theorem guarantees that in bipartite graphs, and thus 3-regular bipartite planar graphs, the size of a minimum vertex cover equals the size of a maximum matching. The classic LP duality for a graph $G=(V,E)$ has:

Primal (maximum matching): $\max \sum_{e \in E} x_e$ subject to $\sum_{e:v \in e} x_e \leq 1$ for all $v$ , $x_e \geq 0$ .
Dual (minimum vertex cover): $\min \sum_{v \in V} y_v$ subject to $y_u + y_v \geq 1$ for all $uv \in E$ , $y_v \geq 0$ .

In bipartite planar graphs, the LP relaxation is exact and the problem is in P; every minimum vertex cover corresponds precisely to a matching of the same cardinality (Kashaev et al., 2022).

For specific families such as $B_{2t}$ (even number of regions) (Imbesi et al., 2012), the vertex covering number (the height of the edge ideal) is given by: $\alpha_0(B_{2t}) = \begin{cases} \frac{3}{4}r + \frac{3}{2} & \text{if } t \text{ is odd} \ \frac{3}{4}r + 1 & \text{if } t \text{ is even} \end{cases}$ where $r = 2t$ is the number of regions.

3. Complexity, Approximation, and Distributed Algorithms

While exact computation is tractable centrally, distributed computation reveals critical limitations. In the LOCAL and CONGEST models:

Maximum matching in bounded-degree, 2-colored graphs admits $(1+\epsilon)$ -approximations in constant time (Göös et al., 2012).
Minimum vertex cover (the dual) is fundamentally harder: no randomized distributed algorithm can compute a $(1+\delta)$ -approximation of minimum vertex cover in $o(\log n)$ time, even on 3-regular bipartite planar graphs.

The lower bound relies on the necessity to coordinate global boundary decisions—with boundary penalties $\partial \ell_o \leq 4\epsilon$ —making vertex cover inherently non-local. In practice, network decomposition and clustering strategies do not overcome this obstacle in strictly local distributed settings (Faour et al., 2020).

For partial cover variants, the Marginally Nonincreasing Coverage (MNC) property holds in bipartite graphs with degree at most 3, yielding polynomial-time algorithms for covering a prescribed number of edges (Caskurlu et al., 2013).

4. Algebraic and Statistical Aspects

Vertex cover properties of $B_{2t}$ graphs are tightly coupled to algebraic invariants:

Graded Betti numbers and projective dimensions provide fine-grained bounds for the minimal free resolution of the edge ideal $I$ $I$ associated to $B_{2t}$ $B_{2 t}$ :
- For $t$ even, $b_{ij}(B_{2t}) \leq \sum_{k+\ell = i+1} \binom{\frac{3t+6}{2}}{k} \binom{\frac{3t+2}{2}}{l}$ .
- Projective dimension satisfies $(3/4)r + 1 < \mathrm{pd}_R(R/I) \leq (3/2)r + 3$ .
- Minimal vertex covers correspond to minimal prime ideals in the primary decomposition (Imbesi et al., 2012).

From a statistical mechanics viewpoint, the solution space of vertex covers is described by Kőnig's theorem and the structure of the unfrozen core:

Replica symmetry can break in sufficiently dense instances, causing the set of backbone vertices to fluctuate (Wei et al., 2015).
Clustering entropy of the unfrozen core can be computed using cycle simplification, reducing cycles to pairs of representative nodes without changing the solution count.

5. Parameterized Complexity and Reconfiguration

The reconfiguration problem for vertex covers (VCR)—transforming one vertex cover into another by a sequence of additions/removals, maintaining cover size at most $k$ —is:

W[1]-hard on bipartite graphs, even though the classic vertex cover problem is in P (Mouawad et al., 2014).
Fixed-parameter tractable on regular (bounded-degree) graphs and nowhere-dense graphs.
NP-hard in general; polynomial-time only for special classes like trees or cactus graphs. For independent set reconfiguration, the analogous problem is PSPACE-complete even when the classic decision problem is tractable.

6. Combinatorial Approximation Schemes

For maximum $k$ -vertex cover in bipartite graphs, a purely combinatorial algorithm achieves an approximation ratio of $0.7$ (Paschos, 2015):

The algorithm "guesses" an optimal split $(k_1, k_2)$ of the $k$ vertices between partitions $V_1$ and $V_2$ , builds candidate solutions, and combines them via parameter elimination.
In strictly 3-regular graphs, degree-based ordering loses effectiveness; one must instead leverage planarity and global structural properties (e.g., separator-based clustering). The extension of multiple-candidate combinatorial schemes to planar 3-regular bipartite graphs is plausible but requires adaptation to uniform degrees and planar constraints.

7. Connections to Enumeration and #P-Hardness

Counting minimum vertex covers on 3-regular bipartite planar graphs is #P-complete (Chambers et al., 6 Oct 2025):

The counting problem reduces to counting noncrossing tilings and then to counting planar straight-line triangulations with fixed cardinal degrees.
In the planar triangulation context, specifying the number of neighbors in each cardinal direction for every vertex does not trivialize the problem; enumeration remains #P-hard.
The reduction chain preserves solution counts, prohibiting efficient enumeration unless $\#\mathbf{P} = \mathbf{FP}$ .

Key formulas in the reduction: $|E| = 3|V| - c - 3$ for maximal triangulations, and the signature saturation for cardinal degrees: $m_\sigma = \sum_{v \in V} \deg_\mathcal{D}(v) = 3|V| - c - 3 \text{ for all } \mathcal{D}$

8. Extremal Cases and Geometric Methods

For broader families of cubic plane graphs (3-regular, 3-connected, and planar), one can remove at most $\sqrt{(p + 3t)n/5}$ edges to eliminate all odd cycles and render the graph bipartite (Nicodemos et al., 2017):

For fullerenes ( $t=0, p=12$ ), the bound strengthens to $\sqrt{12n/5}$ .
Extremal graphs achieving equality have all faces of size 5 or 6, $n=60k^2$ , and icosahedral symmetry.
These bounds yield sharp lower estimates for the size of the maximum cut ( $3n/2 - \sqrt{(p + 3t)n/5}$ ) and maximum independent set ( $n/2 - \sqrt{(p+3t)n/20}$ ).

9. Open Questions and Practical Implications

The key open directions concern distributed approximation limits (matching the $\Omega(\log n)$ barrier), extension of combinatorial schemes to regular and planar cases, development of efficient parameterized algorithms in restricted classes, and deeper algebraic-statistical analysis of solution spaces, especially for large or dense planar graphs.

Applications span computational geometry (e.g., triangulation enumeration), network design (vertex cover in planar communication grids), commutative algebra (Betti number and projective dimension computation), and complexity theory (characterizing #P-hard planar structures).

A plausible implication is that even with strong structural constraints such as planarity, bipartiteness, and regularity, vertex cover problems retain global complexity in both the algorithmic and enumeration senses; local approaches fail to yield highly efficient or strictly local algorithms beyond centralized settings.

References:

(Göös et al., 2012) No Sublogarithmic-time Approximation Scheme for Bipartite Vertex Cover (Imbesi et al., 2012) Algebraic and geometric aspects of bipartite planar graphs (Caskurlu et al., 2013) On Partial Vertex Cover on Bipartite Graphs and Trees (Mouawad et al., 2014) Vertex Cover Reconfiguration and Beyond (Paschos, 2015) Combinatorial approximation of maximum $k$ -vertex cover in bipartite graphs within ratio~0.7 (Wei et al., 2015) Research on Solution Space of Bipartite Graph Vertex-Cover by Maximum Matchings (Nicodemos et al., 2017) Packing and covering odd cycles in cubic plane graphs with small faces (Adams et al., 2018) On planar bipartite biregular degree sequences (Faour et al., 2020) Approximate Bipartite Vertex Cover in the CONGEST Model (Kashaev et al., 2022) Round and Bipartize for Vertex Cover Approximation (Chambers et al., 6 Oct 2025) Counting Triangulations of Fixed Cardinal Degrees