Endpoint extendable paths in dense graphs

A path in a graph is called extendable if it is a proper subpath of another path. A graph is locally connected   if every neighborhood induces a connected subgraph. We show that, for each graph GG of order nn, there exists a threshold number ss such that every path of order smaller than ss is extendable and there exists a non-extendable path of order tt for each t∈{s,…,n−1}t∈{s,…,n−1} if GG satisfies any one of the following three conditions:
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• the degree sum d(u)+d(v)≥nd(u)+d(v)≥n for any two nonadjacent vertices uu and vv;
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• P4P4-free and ω(G−S)≤|S|ω(G−S)≤|S| for every cut set SS of GG;
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• connected, locally connected, and K1,3K1,3-free where P4P4 and K1,3K1,3 denote a path of order 44 and a complete bipartite graph with 11 and 33 vertices in each color class, respectively.