Extremal graphs of the pth power of paths

An extremal graph for a given graph H is a graph with the maximum number of edges among all H-free graphs with fixed number of vertices. The pth power of a path is a graph obtained from a path by joining all pairs of vertices with distance at most p. Applying a deep theorem of Simonovits, we characterize the extremal graphs for the pth power of paths. This settles a conjecture posed by Xiao, Katona, Xiao and Zamora in a stronger form.