Path eccentricity of graphs

Let G be a connected graph. The eccentricity of path P, denoted by eccG(P), is the maximum distance from P to any vertex in G. In Central (CP) problem, our aim find minimum eccentricity. This problem was introduced Cockayne et al., 1981, study different centrality measures on graphs. They showed that CP can solved linear time trees, but it known NP-hard many classes graphs such as chordal bipartite graphs, planar 3-connected split etc. We investigate graph parameter. pe(G) denote value eccG(P) for central obtain tight upper bounds some classes. show pe(G)≤1 biconvex and design an algorithm finds time. On other hand, investigating longest paths graph, we arbitrary k-connected Finally, relation between that, permutation also path. Furthermore, class exhibit superclass does not satisfy this property.