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41.
B.P. Tan 《Discrete Mathematics》2008,308(12):2564-2570
Reid [Every vertex a king, Discrete Math. 38 (1982) 93-98] showed that a non-trivial tournament H is contained in a tournament whose 2-kings are exactly the vertices of H if and only if H contains no transmitter. Let T be a semicomplete multipartite digraph with no transmitters and let Kr(T) denote the set of r-kings of T. Let Q be the subdigraph of T induced by K4(T). Very recently, Tan [On the kings and kings-of-kings in semicomplete multipartite digraphs, Discrete Math. 290 (2005) 249-258] proved that Q contains no transmitters and gave an example to show that the direct extension of Reid's result to semicomplete multipartite digraphs with 2-kings replaced by 4-kings is not true. In this paper, we (1) characterize all semicomplete digraphs D which are contained in a semicomplete multipartite digraph whose 4-kings are exactly the vertices of D. While it is trivial that K4(Q)⊆K4(T), Tan [On the kings and kings-of-kings in semicomplete multipartite digraphs, Discrete Math. 290 (2005) 249-258] showed that K3(Q)⊆K3(T) and K2(Q)=K2(T). Tan [On the kings and kings-of-kings in semicomplete multipartite digraphs, Discrete Math. 290 (2005) 249-258] also provided an example to show that K3(Q) need not be the same as K3(T) in general and posed the problem: characterize all those semicomplete multipartite digraphs T such that K3(Q)=K3(T). In the course of proving our result (1), we (2) show that K3(Q)=K3(T) for all semicomplete multipartite digraphs T with no transmitters such that Q is a semicomplete digraph. 相似文献
42.
Chung-wu Ho 《Proceedings of the American Mathematical Society》2002,130(9):2625-2630
In a conference on fixed point theory, B. Halpern of Indiana University considered the problem of reducing the number of periodic points of a map by homotopy. He also asked whether the number of periodic points of a function could be increased by a homotopy. In this paper, we will show that for any map on a closed manifold, an arbitrarily small perturbation can always create infinitely many periodic points of arbitrarily high periods.
43.
S. P. Strunkov 《Mathematical Notes》2006,80(3-4):590-592
We prove that any set of pair-wise nonisomorphic strongly connected weakly cospectral pseudodigraphs whose set of nilpotency indices is finite also is finite. 相似文献
44.
In this paper, we present a linear time algorithm for constructing linkless drawings of series parallel digraphs with maximum number of symmetries. Linkless drawing in three dimensions is a natural extension to planar drawing in two dimensions. Symmetry is one of the most important aesthetic criteria in graph drawing. More specifically, we present two algorithms: a symmetry finding algorithm which finds maximum number of three dimensional symmetries, and a drawing algorithm which constructs linkless symmetric drawings of series parallel digraphs in three dimensions. 相似文献
45.
A vertex x in a digraph D is said to resolve a pair u, v of vertices of D if the distance from u to x does not equal the distance from v to x. A set S of vertices of D is a resolving set for D if every pair of vertices of D is resolved by some vertex of S. The smallest cardinality of a resolving set for D, denoted by dim(D), is called the metric dimension for D. Sharp upper and lower bounds for the metric dimension of the Cayley digraphs Cay(Δ:Γ), where Γ is the group Zn1⊕Zn2⊕?⊕Znm and Δ is the canonical set of generators, are established. The exact value for the metric dimension of Cay({(0,1),(1,0)}:Zn⊕Zm) is found. Moreover, the metric dimension of the Cayley digraph of the dihedral group Dn of order 2n with a minimum set of generators is established. The metric dimension of a (di)graph is formulated as an integer programme. The corresponding linear programming formulation naturally gives rise to a fractional version of the metric dimension of a (di)graph. The fractional dual implies an integer dual for the metric dimension of a (di)graph which is referred to as the metric independence of the (di)graph. The metric independence of a (di)graph is the maximum number of pairs of vertices such that no two pairs are resolved by the same vertex. The metric independence of the n-cube and the Cayley digraph Cay(Δ:Dn), where Δ is a minimum set of generators for Dn, are established. 相似文献
46.
47.
关于有向整谱图的若干存在性问题 总被引:1,自引:0,他引:1
系统的回答了[1]在1974年提出的关于有向整谱图的两个问题:1.什么样的非对称强连通有向图是整的?是否存在这样的有向图?2.什么样的多重有向图是整的?。 相似文献
48.
设G=(V,A)是一个有向图,其中V和A分别表示有向图G的点集和弧集.对集合TV(G),如果对于任意点v∈V(G)\T,都存在点u,w∈T(u,w可能是同一点)使得(u,v),(v,w)∈A(G),则称T是G的一个双向控制集.有向图G的双向控制数γ~*(G)是G的最小双向控制集所含点的数目.提出了广义de Bruijn和Kautz有向图的双向控制数的新上界,改进了以前文献中提出的相关结论.此外,对某些特殊的广义de Bruijn和Kautz有向图,通过构造其双向控制集,进一步改进了它们双向控制数的上、下界. 相似文献
49.
有向图D=(V, E)被称为是极大弧连通的,如果λ(D)=δ(D)。此外,有向图D被称为是超弧连通的,如果每个最小的弧割都是其某个点的入弧集或者出弧集。以X1和X2为两部的一个有向二部图是半传递的,如果自同构群Aut(D)分别传递的作用在X1和X2上。在这篇论文中,证明了强连通的半传递有向图是极大弧连通的。还证明了除了少部分例外,连通半传递平衡有向图是超弧连通的。 相似文献
50.
An edge cut of a connected graph is m-restricted if its removal leaves every component having order at least m. The size of minimum m-restricted edge cuts of a graph G is called its m-restricted edge connectivity. It is known that when m≤4, networks with maximal m-restricted edge connectivity are most locally reliable. The undirected binary Kautz graph UK(2,n) is proved to be maximal 2- and 3-restricted edge connected when n≥3 in this work. Furthermore, every minimum 2-restricted edge cut disconnects this graph into two components, one of which being an isolated edge. 相似文献