### Abstract

Let G be a graph. The connectivity of G, kappa(G), is the maximum integer k such that there exists a k-container between any two different vertices. A k-container of G between u and v, C(k)(u, v), is a set of k-internally-disjoint paths between u and v. A spanning container is a container that spans V(G). A graph G is k*-connected if there exists a spanning k-container between any two different vertices. The spanning connectivity of G, kappa*(G), is the maximum integer k such that G is w*-connected for 1 <= w <= k if G is 1*-connected.
Let x be a vertex in G and let U = {y(1), y(2,) . . . , y(d)} be a subset of V(G) where x is not in U. A spanning k - (x, U)-fan, F(k)(x, U), is a set of internally-disjoint paths {P(1), P(2,) . . . , P(k)} such that P(i) is a path connecting x to y(i) for 1 <= i <= k and U(i=1)(k) V(P(i)) = V(G). A graph G is k*-fan-connected (or k(f)*-connected) if there exists a spanning Fk(x, U)-fan for every choice of x and U with vertical bar U vertical bar = k and x is not an element of U. The spanning fan-connectivity of a graph G, kappa(f)*(G), is defined as the largest integer k such that G is w(f)*-connected for 1 <= w <= k if G is 1(f)*-connected.
In this paper, some relationship between kappa(G), kappa*(C), and kappa(f)*(G) are discussed. Moreover, some sufficient conditions for a graph to be k(f)*-connected are presented. Furthermore, we introduce the concept of a spanning pipeline-connectivity and discuss some sufficient conditions for a graph to be k*-pipeline-connected. Published by Elsevier B.V.

Original language | English |
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Pages (from-to) | 1342-1348 |

Number of pages | 17 |

Journal | Discrete Applied Mathematics |

Volume | 157 |

Issue number | 7 |

DOIs | |

State | Published - 6 Apr 2009 |

### Keywords

- Hamiltonian connected; Hamiltonian; Dirac Theorem; Menger Theorem; Ore Theorem; Connectivity; Spanning connectivity; Spanning fan-connectivity; Spanning pipeline-connectivity; Graph container

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## Cite this

Lin, C-K., Tan, J-M., Frank Hsu, D., & Hsu, L-H. (2009). On the spanning fan-connectivity of graphs.

*Discrete Applied Mathematics*,*157*(7), 1342-1348. https://doi.org/10.1016/j.dam.2008.11.014