A new family of proximity graphs: Class cover catch digraphs

Motivated by issues in machine learning and statistical pattern classification, we investigate a class cover problem (CCP) with an associated family of directed graphs—class cover catch digraphs (CCCDs). CCCDs are a special case of catch digraphs. Solving the underlying CCP is equivalent to finding a smallest cardinality dominating set for the associated CCCD, which in turn provides regularization for statistical pattern classification. Some relevant properties of CCCDs are studied and a characterization of a family of CCCDs is given.     

非负矩阵与有向图的谱半径

本文给出非负矩阵的谱半径的上界、下界,由此给出有向图的谱半径的界.     

Directed triangles in directed graphs

We show that each directed graph (with no parallel arcs) on n vertices, each with indegree and outdegree at least n/twhere t=2.888997… contains a directed circuit of length at most 3.     

A note on theβ-measure for digraph competitions

Digraph games are cooperative TU-games associated to digraph competitions: domination structures that can be modeled by directed graphs. Examples come from sports competitions or from simple majority win digraphs corresponding to preference profiles for a group of individuals within the framework of social choice theory. Brink and Gilles (2000) defined theβ-measure of a digraph competition as the Shapley value of the corresponding digraph game. This paper provides a new characterization of theβ-measure.     

Cyclical games with prohibitions

We consider a certain combinatorial game on a digraph for two cases of the price function. For one case the game in question extends the cyclical game studied in Ehrenfeucht and Mycielski (1979) and Gurvitch, Karzanov and Khachiyan (1988) which, in its turn, is a generalization of the well-known problem of finding a minimum mean cycle in an edge-weighted digraph. We prove the existence of optimal uniform stationary strategies for both cases and give algorithms to find such strategies.This work was partially supported by Chaire municipale, Mairie de Grenoble.     

Extremal digraphs avoiding distinct walks of length 3 with the same endpoints

In this paper, we determine the maximum size of digraphs on n vertices which avoid two distinct walks of length 3 with the same initial vertex and the same terminal vertex. The digraphs attaining this maximum size are also characterized. Combining this with previous results, we obtain a full solution to a problem proposed by X. Zhan in 2007.     

2—有向循环图的同构

本文得到了2-有向循环图 G(a,b,N)与 G(a＇,b＇,N)同构的充要条件为:(a,b,N)=(a＇,b＇,N)=K且 a＇b≡ab＇(mod KN);或 a＇a≡b＇b(mod KN),这里(x,y,z)表示整数 x,y,z 的最大公约数。     

Approximation algorithms for constructing some required structures in digraphs

We consider a new problem of constructing some required structures in digraphs, where all arcs installed in such required structures are supposed to be cut from some pieces of a specific material of length L. Formally, we consider the model: a digraph D = (V, A; w), a structure S and a specific material of length L, where w: A → R⁺, we are asked to construct a subdigraph D′ from D, having the structure S, such that each arc in D′ is constructed by a part of a piece or/and some whole pieces of such a specific material, the objective is to minimize the number of pieces of such a specific material to construct all arcs in D′.     

<Emphasis Type="Bold">R</Emphasis><Emphasis Type="SmallCaps">UBIS</Emphasis>: a bipolar-valued outranking method for the choice problem

The main concern of this article is to present the R UBIS method for tackling the choice problem in the context of multiple criteria decision aiding. Its genuine purpose is to help a decision maker to determine a single best decision alternative. Methodologically we focus on pairwise comparisons of these alternatives which lead to the concept of bipolar-valued outranking digraph. The work is centred around a set of five pragmatic principles which are required in the context of a progressive decision aiding methodology. Their thorough study and implementation in the outranking digraph lead us to define a choice recommendation as an extension of the classical digraph kernel concept.      

Characterizations of Clifford semigroup digraphs

This paper characterizes directed graphs which are Cayley graphs of strong semilattices of groups and, in particular, strong chains of groups, i.e. of completely regular semigroups which are also called Clifford semigroups.