Endpoint extendable paths in dense graphs

Let G be a graph of order n. A path P of G is extendible if it can be extended to a longer path from one of its two endvertices, otherwise we say P is non-extendible. Let G be a graph of order n. We show that there exists a threshold number s such that every path of order smaller than s is extendible and there exists a non-extendible path of order t for each t ∈ {s, s+ 1, · · · , n} provided G satisfies one of the following three conditions: • d(u) + d(v) ≥ n for any two of nonadjacent vertices u and v. • G is a P4-free 1-tough graph. • G is a connected, locally connected, and K1,3-free graph.