The claimed QMA1-hardness of sparse balancedness and sparse bipartitedness is not established, because the main spectral equivalence is false.
Zaslavsky, Negative (and positive) circles in signed gra phs: A problem collection, AKCE International Journal of Gr aphs and Combinatorics 15, 31 (2018)
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Testing the presence of balanced and bipartite components in a sparse graph is QMA1-hard
The claimed QMA1-hardness of sparse balancedness and sparse bipartitedness is not established, because the main spectral equivalence is false.