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We define a total order, which we call rooted order, on minimal generatingset of $J(P_n)^s$ where $J(P_n)$ is the cover ideal of a path graph on $n$vertices. We show that each power of a cover ideal of a path has linearquotients with respect to the rooted order. Along the way, we characterizeminimal generating set of $J(P_n)^s$ for $s\geq 3$ in terms of minimalgenerating set of $J(P_n)^2$. We also discuss the extension of the concept ofrooted order to chordal graphs. Computational examples suggest that such ordergives linear quotients for powers of cover ideals of chordal graphs as well.

Rooted order on minimal generators of powers of some cover ideals