A k-tree of a connected graph G is a spanning tree with maximum degree at most k. The rupture degree for a connected graph G is defined by , where and , respectively, denote the order of the largest component and numb...A k-tree of a connected graph G is a spanning tree with maximum degree at most k. The rupture degree for a connected graph G is defined by , where and , respectively, denote the order of the largest component and number of components in . In this paper, we show that for a connected graph G, if for any cut-set , then G has a k-tree.展开更多
We employ graph parameter, the rupture degree, to measure the vulnerability of <em>k</em>-uniform hypergraph <em>G<sup>k</sup></em>. For the k-uniform hypergraph <em>G<sup&...We employ graph parameter, the rupture degree, to measure the vulnerability of <em>k</em>-uniform hypergraph <em>G<sup>k</sup></em>. For the k-uniform hypergraph <em>G<sup>k</sup></em> underlying a non-complete graph <em>G</em> = (<em>V</em>, <em>E</em>), its rupture degree <em>r</em>(<em>G<sup>k</sup></em>) is defined as <em>r</em>(<em>G<sup>k</sup></em>) = max{<em>ω</em>(<em>G<sup>k</sup></em> - <em>X</em>) - |<em>X</em>| - <em>m</em>(<em>G<sup>k</sup></em> - <em>X</em>): <em>X</em> <span style="white-space:nowrap;">⊂</span> <em>V</em>(<em>G<sup>k</sup></em>), <em>ω</em>(<em>G<sup>k</sup></em> - <em>X</em>) > 1}, where <em>X</em> is a cut set (or destruction strategy) of <em>G<sup>k</sup></em>, <em>ω</em>(<em>G<sup>k</sup></em> - <em>X</em>) and <em>m</em>(<em>G<sup>k</sup></em> - <em>X</em>) denote the number of components and the order of a largest component in <em>G<sup>k</sup></em> - <em>X</em>, respectively. It is shown that this parameter can be used to measure the vulnerability of networks. In this paper, the rupture degrees of several specific classes of <em>k</em>-uniform hypergraph are determined.展开更多
The rupture degree of a noncomplete-connected graph G is defined by , where is the number of components of and is the order of the largest component of. In this paper, we determine the rupture degree of some Cartesian...The rupture degree of a noncomplete-connected graph G is defined by , where is the number of components of and is the order of the largest component of. In this paper, we determine the rupture degree of some Cartesian product graphs.展开更多
文摘A k-tree of a connected graph G is a spanning tree with maximum degree at most k. The rupture degree for a connected graph G is defined by , where and , respectively, denote the order of the largest component and number of components in . In this paper, we show that for a connected graph G, if for any cut-set , then G has a k-tree.
文摘We employ graph parameter, the rupture degree, to measure the vulnerability of <em>k</em>-uniform hypergraph <em>G<sup>k</sup></em>. For the k-uniform hypergraph <em>G<sup>k</sup></em> underlying a non-complete graph <em>G</em> = (<em>V</em>, <em>E</em>), its rupture degree <em>r</em>(<em>G<sup>k</sup></em>) is defined as <em>r</em>(<em>G<sup>k</sup></em>) = max{<em>ω</em>(<em>G<sup>k</sup></em> - <em>X</em>) - |<em>X</em>| - <em>m</em>(<em>G<sup>k</sup></em> - <em>X</em>): <em>X</em> <span style="white-space:nowrap;">⊂</span> <em>V</em>(<em>G<sup>k</sup></em>), <em>ω</em>(<em>G<sup>k</sup></em> - <em>X</em>) > 1}, where <em>X</em> is a cut set (or destruction strategy) of <em>G<sup>k</sup></em>, <em>ω</em>(<em>G<sup>k</sup></em> - <em>X</em>) and <em>m</em>(<em>G<sup>k</sup></em> - <em>X</em>) denote the number of components and the order of a largest component in <em>G<sup>k</sup></em> - <em>X</em>, respectively. It is shown that this parameter can be used to measure the vulnerability of networks. In this paper, the rupture degrees of several specific classes of <em>k</em>-uniform hypergraph are determined.
文摘The rupture degree of a noncomplete-connected graph G is defined by , where is the number of components of and is the order of the largest component of. In this paper, we determine the rupture degree of some Cartesian product graphs.
