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Abstract
Let G be a connected graph of order p ≥ 2. For any vertex x in G, a set S of vertices of G is an x-detour set if each vertex v of G lies on an x-y detour in G for some element y in S. The minimum cardinality of an x-detour set of G is defined as the x-detour number of G, denoted by dx(G) or simply dx. An x-detour set of cardinality dx (G) is called a dx-set of G. We determine bounds for it and characterize graphs which realize these bounds. We define an x-superior vertex of a graph and characterize graphs G for which dx(G) = 1 in terms of x-superior vertices. Also, we find its relation with the detour number of a graph. It is shown that if G is a graph of order p, then dx(G) ≤ p-eD(x) for any vertex x in G. Connected graphs of order p with vertex detour numbers p-l or p-2 for every vertex are characterized. For positive integers R, D and n ≥ 2 with R < D ≤ 2R, there exists a connected graph G with radD G = R, diamD G = D and dx(G) = n or dx(G) = n-1 for every vertex x of G. For each triple D, n and p of integers with 1 ≤ n ≤ p - D + 1 and D ≥ 4, there is a connected graph G of order p, detour diameter D and dx(G) = n or dx(G) = n-1 for every vertex x of G. Also, for an integer p ≥ 2 and a number n with 1 ≤ n ≤ p - 1, there exists a connected graph G of order p and dx(G) = n or dx(G) = n - 1 for every vertex x of G.