An Iterative Procedure for Evaluating Digraph Competitions

A competition which is based on the results of (partial) pairwise comparisons can be modelled by means of a directed graph. Given initial weights on the nodes in such digraph competitions, we view the measurement of the importance (i.e., the cardinal ranking) of the nodes as an allocation problem where we redistribute the initial weights on the basis of insights from cooperative game theory. After describing the resulting procedure of redistributing the initial weights, an iterative process is described that repeats this procedure: at each step the allocation obtained in the previous step determines the new input weights. Existence and uniqueness of the limit is established for arbitrary digraphs. Applications to the evaluation of, e.g., sport competitions and paired comparison experiments are discussed.     

Disjoint Paths in Decomposable Digraphs

The k‐linkage problem is as follows: given a digraph and a collection of k terminal pairs such that all these vertices are distinct; decide whether D has a collection of vertex disjoint paths such that is from to for . A digraph is k‐linked if it has a k‐linkage for every choice of 2k distinct vertices and every choice of k pairs as above. The k‐linkage problem is NP‐complete already for [11] and there exists no function such that every ‐strong digraph has a k‐linkage for every choice of 2k distinct vertices of D [17]. Recently, Chudnovsky et al. [9] gave a polynomial algorithm for the k‐linkage problem for any fixed k in (a generalization of) semicomplete multipartite digraphs. In this article, we use their result as well as the classical polynomial algorithm for the case of acyclic digraphs by Fortune et al. [11] to develop polynomial algorithms for the k‐linkage problem in locally semicomplete digraphs and several classes of decomposable digraphs, including quasi‐transitive digraphs and directed cographs. We also prove that the necessary condition of being ‐strong is also sufficient for round‐decomposable digraphs to be k‐linked, obtaining thus a best possible bound that improves a previous one of . Finally we settle a conjecture from [3] by proving that every 5‐strong locally semicomplete digraph is 2‐linked. This bound is also best possible (already for tournaments) [1].     

一类含有奇数个顶点的三色有向图本原指数上界

一个三色有向图D是本原的,当且仅当存在非负整数h、k和v,且h+k+v0,使得D中的每一对顶点(i,j)都存在从i到j的(h,k,v)-途径,h+k+v的最小值定义为三色有向图D的本原指数.研究了一类三色有向图,它的未着色图中包含2佗-4个顶点,一个n-圈、一个(n-2)-圈和一个2-圈,给出了本原指数上界.     

本原极小强连通有向图1—指数的下界

本文给出了n阶本原极小强连通有向图1-指数的下图:expD(1)4.且这个下界是可以达到的.     

在有向网络中寻找最小支撑入树的计算方法

本文研究了有向网络中支撑入树的性质 ,提出了在有向网络图中寻找以某一指定点为根的最小支撑入树的一种较简便的算法 ,并给出了应用该算法的一个实际算例     

线有向图的幂敛指数

设G是有向图，LG表示c的线有向图，本文得到了它们的幂敛指数及周期之间的关系：k（G）－1≤（LG）≤k（G）＋1,p（LG）=p（G）特别地，当G为本原图时有：k（LG）＝k（G）＋1     

对称本原矩阵指数集的刻画

设Ｓｎ表示由全体ｎ阶对称本原（０，１）－矩所构成的集合，并设Ｓ（ｎ，ｄ）＝｛Ａ∈Ｓｎ│Ａ的伴随有向图中的最小奇圈之长为ｄ≥１｝。本文证明了：Ｓ（ｎ，ｄ）的本原指数集为｛ｄ－１，ｄ，…，２ｎ－ｄ－１｝＼Ｄ，其中Ｄ为｛ｎ－ｄ＋１，ｎ－ｄ＋２，…，２ｎ－ｄ－２｝中的所有奇数与０之并集，同时，我们也给出了Ｓ（ｎ，ｄ）中指数达到上界的矩阵集合的完全刻画。     

Partitioning series-parallel multigraphs intoν*-excluding edge covers

We prove that, for any given vertexν* in a series-parallel graph G, its edge set can be partitioned into k= min{k′(G) 1,δ(G)} subsets such that each subset covers all the vertices of G possibly except forν*, whereδ(G) is the minimum degree of G and k′(G) is the edge-connectivity of G. In addition, we show that the results in this paper are best possible and a polynomial time algorithm can be obtained for actually finding such a partition by our proof.     

CONNECTIVITIES OF MINIMAL CAYLEY COSET DIGRAPHS

We prove that the connectivities of minimal Cayley coset digraphs are their regular degrees.