Abstract
In this paper, we introduce the concepts of minimal path cover sets and the index of covering of a simple graph G, and study the basic properties of these notions. Corresponding to these combinatorial concepts, we define max-path ideal and path cover ideal attached to G and study its algebraic properties in the case that G is a tree. Especially, we characterize the trees in which the max-path ideals have height 1 or 2.
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