Abstract
A subset S of vertices of a graph G is called a k - path vertex cover if every path of order k in G contains at least one vertex from S . Denote by ψ k ( G ) the minimum cardinality of a k -path vertex cover in G . It is shown that the problem of determining ψ k ( G ) is NP-hard for each k ≥ 2 , while for trees the problem can be solved in linear time. We investigate upper bounds on the value of ψ k ( G ) and provide several estimations and exact values of ψ k ( G ) . We also prove that ψ 3 ( G ) ≤ ( 2 n + m ) / 6 , for every graph G with n vertices and m edges.
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