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The connected vertex detour number of a graph. (English) Zbl 1235.05045

Summary: For a connected graph G of order p2 and a vertex x of G, a set SV(G) is an x-detour set of G if each vertex vV(G) lies on an x-y detour 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 d x (G). An x-detour set of cardinality d x (G) is called a d x -set of G. A connected x-detour set of G is an x-detour set S such that the subgraph G[S] induced by S is connected.

The minimum cardinality of a connected x-detour set of G is defined as the connected x-detour number of G and is denoted by cd+x(G). A connected x-detour set of cardinality cd x (G) is called a cd x -set of G. We determine bounds for the connected x-detour number and find the same for some special classes of graphs. If a, b and c are positive integers such that 3ab+1<c, then there exists a connected graph G with detour number dn(G)=a, d x (G)=b and cd x (G)=c for some vertex x in G.

For positive integers R, D and n3 with R<D<2R, there exists a connected graph G with rad D G=R, diam D G=D and cd x (G)=n for some vertex x in G. Also, for each triple D, n and p of integers with 4Dp-1 and 3np, there is a connected graph G of order p, detour diameter D and cd x (G)=n for some vertex x of G.

MSC:
05C12Distance in graphs
05C40Connectivity