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We show that attaching a whisker (or a pendant) at the vertices of a cyclecover of a graph results in a new graph with the following property: allsymbolic powers of its cover ideal are Koszul or, equivalently, componentwiselinear. This extends previous work where the whiskers were added to allvertices or to the vertices of a vertex cover of the graph.

Symbolic Powers of Cover Ideals of Graphs and Koszul Property