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The simple greedy algorithm to find a maximal independent set of a graph canbe viewed as a sequential update of a Boolean network, where the updatefunction at each vertex is the conjunction of all the negated variables in itsneighbourhood. In general, the convergence of the so-called kernel network iscomplex. A word (sequence of vertices) fixes the kernel network if applying theupdates sequentially according to that word. We prove that determining whethera word fixes the kernel network is coNP-complete. We also consider theso-called permis, which are permutation words that fix the kernel network. Weexhibit large classes of graphs that have a permis, but we also construct manygraphs without a permis.

Words fixing the kernel network and maximum independent sets in graphs