On the Complexity of Bipartite Degree Realizability
István Miklós
TL;DR
This work investigates the complexity of the Bipartite Degree Realization problem (BDR) without a prescribed vertex partition by introducing the parametric variant BDR_{c1,c2}. It proves polynomial-time decidability for a dense range of degree sequences where c1 and c2 satisfy explicit bounds, using a reduction to least-balanced degree sequences and a targeted Gale–Ryser verification complemented by bounded subset-sum techniques. The authors also establish a conditional NP-hardness dichotomy: if the unrestricted BDR is NP-complete, then BDR_{c1,c2} remains NP-complete for parameter ranges with 0 < c2 < 1/2 and c1 < 1 − c2 − √(1−2c2). Together, these results delineate tractable and hard regions in the space of bipartite realizability for dense degree sequences and connect to broader questions about potentially bipartite graphic degree sequences.
Abstract
We study the \emph{Bipartite Degree Realization} (BDR) problem: given a graphic degree sequence $D$, decide whether it admits a realization as a bipartite graph. While bipartite realizability for a fixed vertex partition can be decided in polynomial time via the Gale--Ryser theorem, the computational complexity of BDR without a prescribed partition remains unresolved. We address this question through a parameterized analysis. For constants $0 \le c_1 \le c_2 \le 1$, we define $\mathrm{BDR}_{c_1,c_2}$ as the restriction of BDR to degree sequences of length $n$ whose degrees lie in the interval $[c_1 n, c_2 n]$. Our main result shows that $\mathrm{BDR}_{c_1,c_2}$ is solvable in polynomial time whenever $0 \le c_1 \le c_2 \le \frac{\sqrt{c_1(c_1+4)}-c_1}{2}$, as well as for all $c_1 > \tfrac12$. The proof relies on a reduction to extremal \emph{least balanced degree sequences} and a detailed verification of the critical Gale--Ryser inequalities, combined with a bounded subset-sum formulation. We further show that, assuming the NP-completeness of unrestricted BDR, the problem $\mathrm{BDR}_{c_1,c_2}$ remains NP-complete for all $0 < c_2 < \tfrac12$ and $c_1 < 1 - c_2 - \sqrt{1-2c_2}$. Our results clarify the algorithmic landscape of bipartite degree realization and contribute to the broader study of potentially bipartite graphic degree sequences.