Minimum Degree up to Local Complementation: Bounds, Parameterized Complexity, and Exact Algorithms

TL;DR

This paper investigates the local minimum degree of graphs, establishing bounds, complexity classifications, and exact algorithms, revealing its computational hardness and providing improved bounds and algorithms for bipartite graphs.

Contribution

It provides new bounds on the local minimum degree, proves NP-completeness and parameterized complexity results, and introduces exact algorithms with improved exponential running times.

Findings

Lower bounds of 0.189n for general graphs and 0.110n for bipartite graphs.

Upper bounds of 3n/8+o(n) for general graphs and n/4+o(n) for bipartite graphs.

NP-Completeness and W[2]-hardness of the problem, with exact algorithms running in O*(1.938^n) and O*(1.466^n) for bipartite graphs.

Abstract

The local minimum degree of a graph is the minimum degree that can be reached by means of local complementation. For any n, there exist graphs of order n which have a local minimum degree at least 0.189n, or at least 0.110n when restricted to bipartite graphs. Regarding the upper bound, we show that for any graph of order n, its local minimum degree is at most 3n/8+o(n) and n/4+o(n) for bipartite graphs, improving the known n/2 upper bound. We also prove that the local minimum degree is smaller than half of the vertex cover number (up to a logarithmic term). The local minimum degree problem is NP-Complete and hard to approximate. We show that this problem, even when restricted to bipartite graphs, is in W[2] and FPT-equivalent to the EvenSet problem, which W[1]-hardness is a long standing open question. Finally, we show that the local minimum degree is computed by a O*(1.938^n)-algorithm, and a O*(1.466^n)-algorithm for the bipartite graphs.