Core and Conditional Core Path of Specified Length in Special Classes of Graphs

A core path of a graph is a path P in G that minimizes d(P ) = ∑ v∈V d(v, P )w(v). In this paper, we study the location of core path of specified length in special classes of graphs. Further, we extend our study to the problem of locating a core path of specified length under the condition that some existing facilities are already located (known as conditional core path of a graph). We study both the problems stated above in vertex weighted bipartite permutation graphs, threshold graphs and proper interval graphs and give polynomial time algorithms for the core path and conditional core path problem in these classes. We also establish the NP-Completeness of the above problems in the same classes of graphs when arbitrary positive weights are assigned to edges.