Proper interval graphs are exactly the graphs whose independence complex is sortable, and this implies strong persistence and linear quotients for powers of t-independence ideals of these graphs.
Gardi, The Roberts characterization of proper and unit interval gr aphs, Discrete Mathe- matics 307, no
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Sortable simplicial complexes and $t$-independence ideals of proper interval graphs
Proper interval graphs are exactly the graphs whose independence complex is sortable, and this implies strong persistence and linear quotients for powers of t-independence ideals of these graphs.