3-path vertex cover and dissociation number of hexagonal graphs

A subset P 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 P. The cardinality of a minimum k-path vertex cover is called the k-path vertex cover number of G, and is denoted by ?k(G). It is known that the problem of finding a minimum 3-path vertex cover is NP-hard for planar graphs. In this paper we establish an upper bound for ?3(G), where G is from an important family of planar graphs, called hexagonal graphs, arising from real world applications.