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A famous example of the periodicity forcing word equation is $xy =
yx$. It means that if a word $xy$ equals to a word $yx$, obtained by
a _cut-and-swap_ process, there exists a word $w$ and integers
$i,j$ such that $x = w^i$ and $y = w^j$. The paper's Section 1 and 5 contain a
brief yet very helpful overview of existing results about periodicity
forcing.
About ten years ago, Czeizler, Kari and Seki, in their work [On a
special class of primitive words](https://www.csd.uwo.ca/~lkari/mfcs.pdf), have introduced a notion of a
pseudo-repetition. It can be explained with the following example.
When DNA molecules are regarded as words, the Watson-Crick
correspondence $$G \sim C, A \sim T$$ between guanine ($G$) and
cytosine ($C$), adenine ($A$) and thymine ($T$) allows us to say that
a word $ATCGATTGAGCTCTAGCG$ contains the same information as the word
$TAGCTAACTCGAGATCGC$. A pseudo-repetition is a repetition of a word
or its Watson-Crick complement. So, for instance $ATC \sim TAG$, and
the word $ATCTAGATCATCTAG$ is a pseudo-repetition of the word $ATC$.
In an abstract algebraic setting (involving monoids, morphic
permutations, equivalence classes and anticongruences), the author
studies an extension to the notion of periodicity forcing, using the
recently introduced concept of pseudo-repetitions and related
ideas. Holub proves, for instance, that any periodicity forcing
classical word equation also forces a pseudo-periodicity. He provides
interesting new results and generalizations of already known results.
MathSciNet:MR4074724.

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Nice coarse-gaining based heuristic to partition the nodes into k sets of similar sizes minimizing the number of edges between those sets.
Random maximal matchings are used to coarse-grain the graph. Maybe fast heuristics to obtain a large maximal matching can be considered, such as using the two rules detailed here: [Karp-Sipser based kernels for bipartite graph matching](https://papers-gamma.link/paper/160).
Publicly available implementation: http://glaros.dtc.umn.edu/gkhome/metis/metis/overview

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