Words fixing the kernel network and maximum independent sets in graphs

TL;DR

This paper investigates the complexity of fixing kernel networks in graphs using specific vertex sequences, proving coNP-completeness and exploring classes of graphs with permutation fixing sequences.

Contribution

It introduces the concept of words fixing kernel networks, proves the problem's coNP-completeness, and characterizes classes of graphs with permutation fixing sequences.

Findings

Determined coNP-completeness of fixing kernel networks.

Identified large classes of graphs with permutation fixing sequences.

Constructed graphs without permutation fixing sequences.

Abstract

The simple greedy algorithm to find a maximal independent set of a graph can be viewed as a sequential update of a Boolean network, where the update function at each vertex is the conjunction of all the negated variables in its neighbourhood. In general, the convergence of the so-called kernel network is complex. A word (sequence of vertices) fixes the kernel network if applying the updates sequentially according to that word. We prove that determining whether a word fixes the kernel network is coNP-complete. We also consider the so-called permis, which are permutation words that fix the kernel network. We exhibit large classes of graphs that have a permis, but we also construct many graphs without a permis.