Degree-constrained 2-partitions of graphs

TL;DR

This paper investigates the computational complexity of partitioning graphs into two parts with specified minimum degree constraints, providing a complete classification and identifying thresholds for NP-completeness.

Contribution

It determines the complexity of degree-constrained 2-partitions for all positive integers and establishes the function g(k1,k2) that delineates polynomial and NP-complete cases.

Findings

Complexity classification for all positive k1,k2

Exact values of g(k1,k2) for specific cases

Identification of thresholds for NP-completeness

Abstract

A $(\delta\geq k_1,\delta\geq k_2)$-partition of a graph $G$ is a vertex-partition $(V_1,V_2)$ of $G$ satisfying that $\delta(G[V_i])\geq k_i$ for $i=1,2$. We determine, for all positive integers $k_1,k_2$, the complexity of deciding whether a given graph has a $(\delta\geq k_1,\delta\geq k_2)$-partition. We also address the problem of finding a function $g(k_1,k_2)$ such that the $(\delta\geq k_1,\delta\geq k_2)$-partition problem is ${\cal NP}$-complete for the class of graphs of minimum degree less than $g(k_1,k_2)$ and polynomial for all graphs with minimum degree at least $g(k_1,k_2)$. We prove that $g(1,k)=k$ for $k\ge 3$, that $g(2,2)=3$ and that $g(2,3)$, if it exists, has value 4 or 5.