完全多部图的区间图完全化问题

The interval graph completion problem of a graph G includes two class problems: the profile problem and the pathwidth problem, denoted as P(G) and PW(G) respectively, where the profile problem is to find an interval supergraph with the smallest possible number of edges; the pathwidth problem is to find an interval supergraph with the smallest possible cliquesize. These two class problems have important applications to numerical algebra, VLSI-layout and algorithm graph theory respectively; And they are known to be NP-complete for general graphs. Some classes of special graphs have been investigated in the literatures. In this paper the exact solutions of the profile and the pathwidth of the complete multipartite Graph Kn1,n2,…,nr(r≥2) are determined.     

关于区间图的一些有效并行算法

本文基于SIMD-CREW-PRAM一种可同时读但不可同时写的共享计算模型,给出了有关区间图的一些有效的并行算法。如找最大加权集团,最小集团覆盖,等权区间图的最大独立集及最小支配集,真区间图的哈密顿回路及最小带宽。所有这些算法都是最优的,其时间复杂性为O(logn)和处理器数为O(n)。     

赋权图的圆染色与区间染色

赋权图的区间染色的定义与赋权图的圆染色的定义非常类型,唯一的区别就是将Ｇ的顶点对应圆周上的孤换为Ｇ的顶点对应区间上的子区间,讨论了赋权的圆染色与区染色的关系。     

区间支付图对策上的平均树解

结合图对策和具有区间支付的模糊合作对策理论,引入区间支付图对策,提出区间平均树解,此解可以看做是经典图对策中平均树解的推广,并通过算例说明区间平均树解的应用性.分析了区间平均树解的相对分支有效性.当区间支付图对策满足严格超可加性时,每个局中人参加联盟得到的收益不小于其单干所得支付.最后,讨论了经典平均树解与区间平均树解之间的关系.     

赋权二部图最大匹配的灵敏分析

本主要从理论上讨论赋权二部图的权的变化对最优解的影响，并在原最大权匹配的基础上给出求解权值变化后的最大权匹配的算法。     

区间值模糊图的运算性质

证明了区间值强模糊图类对笛卡尔积和合成运算封闭但对并运算和联运算不封闭,给出了它们对并运算和联运算封闭的条件,研究了合成、并、联运算和补运算之间的关系.     

二部图的距离k次方和问题

本文定义S_k（G）为G中所有点对之间距离的k次方之和.利用顶点划分的方法得到了直径为d的n顶点连通二部图S_k（G）的下界,并确定了达到下界所对应的的极图.     

可变抽样区间的单边控制图

利用质量控制图监督生产过程时 ,通常每隔固定时间从过程抽取固定容量的样本。本文在前文[1] 的基础上设计具有可变抽样区间的单边标准差 (S)图、极差 (R)图和不合格品数 (np)图。计算了这三个图发信号前的平均样本数和平均时间 ,并同固定抽样区间的常规控制图作比较。所设计的控制图能缩短过程失控时间从而减少不合格品数。     

平衡二部图的四圈覆盖

对任意的正整数ｋ,如果每部２ｋ个点的平衡二部图Ｇ＝（ｖ１,ｖ２;Ｅ）的最小度大于等于４ｋ／３,那么Ｇ恰好被ｋ个相互独立的四圈覆盖。     

OPTIMAL LABELLING OF UNIT INTERVAL GRAPHS

This paper shows that, for every unit interval graph, there is a labelling which is simultaneously optimal for the following seven graph labelling problems: bandwidth, cyclic bandwidth, profile, fill-in, cutwidth, modified cutwidth, and bandwidth sum(linear arrangement).     

Adjacency matrices of probe interval graphs

In this paper we obtain several characterizations of the adjacency matrix of a probe interval graph. In course of this study we describe an easy method of obtaining interval representation of an interval bigraph from its adjacency matrix. Finally, we note that if we add a loop at every probe vertex of a probe interval graph, then the Ferrers dimension of the corresponding symmetric bipartite graph is at most 3.     

Recognizing and representing proper interval graphs in parallel using merging and sorting

We present a parallel algorithm for recognizing and representing a proper interval graph in time with O(m+n) processors on the CREW PRAM, where m and n are the number of edges and vertices in the graph. The algorithm uses sorting to compute a weak linear ordering of the vertices, from which an interval representation is easily obtained. It is simple, uses no complex data structures, and extends ideas from an optimal sequential algorithm for recognizing and representing a proper interval graph [X. Deng, P. Hell, J. Huang, Linear-time representation algorithms for proper circular-arc graphs and proper interval graphs, SIAM J. Comput. 25 (2) (1996) 390-403].     

A dynamic distributed approach to representing proper interval graphs

First studied by Brodal and Fagerberg [G.S. Brodal, R. Fagerberg, Dynamic representation of sparse graphs, in: Algorithms and Data Structures, Proceedings of the 6th International Workshop, Vancouver, Canada, in: Lecture Notes in Computer Science, vol. 1663, Springer-Verlag, 1999], a dynamic adjacency labelling scheme labels the vertices of a graph so that the adjacency of two vertices can be deduced from their labels. The scheme is dynamic in the sense that only a small adjustment must be made to the vertex labels when a small change is made to the graph.Using a centralized dynamic representation of Hell, Shamir and Sharan [P. Hell, R. Shamir, R. Sharan, A fully dynamic algorithm for recognizing and representing proper interval graphs, SIAM Journal on Computing 31 (1) (2001) 289-305], we develop a bit/label dynamic adjacency labelling scheme for proper interval graphs. Our fully dynamic scheme handles vertex deletion/addition and edge deletion/addition in time. Furthermore, our dynamic scheme is error-detecting, as it recognizes when the new graph is not a proper interval graph.     

A characterization of cycle-free unit probe interval graphs

A graph is a probe interval graph (PIG) if its vertices can be partitioned into probes and nonprobes with an interval assigned to each vertex so that vertices are adjacent if and only if their corresponding intervals overlap and at least one of them is a probe. PIGs are a generalization of interval graphs introduced by Zhang for an application concerning the physical mapping of DNA in the human genome project. PIGs have been characterized in the cycle-free case by Sheng, and other miscellaneous results are given by McMorris, Wang, and Zhang. Johnson and Spinrad give a polynomial time recognition algorithm for when the partition of vertices into probes and nonprobes is given. The complexity for the general recognition problem is not known. Here, we restrict attention to the case where all intervals have the same length, that is, we study the unit probe interval graphs and characterize the cycle-free graphs that are unit probe interval graphs via a list of forbidden induced subgraphs.     

Proper interval graphs and the guard problem

This paper is a study of the hamiltonicity of proper interval graphs with applications to the guard problem in spiral polygons. We prove that proper interval graphs with 2 vertices have hamiltonian paths, those with 3 vertices have hamiltonian cycles, and those with 4 vertices are hamiltonian-connected if and only if they are, respectively, 1-, 2-, or 3-connected. We also study the guard problem in spiral polygons by connecting the class of nontrivial connected proper interval graphs with the class of stick-intersection graphs of spiral polygons.     

A maximal zero-free interval for chromatic polynomials of bipartite planar graphs

It is well known that (-∞,0) and (0,1) are two maximal zero-free intervals for all chromatic polynomials. Jackson [A zero-free interval for chromatic polynomials of graphs, Combin. Probab. Comput. 2 (1993), 325-336] discovered that is another maximal zero-free interval for all chromatic polynomials. In this note, we show that is actually a maximal zero-free interval for the chromatic polynomials of bipartite planar graphs.     

Mutual exclusion scheduling with interval graphs or related classes: Complexity and algorithms

This note summarizes the main results presented in the author's Ph.D. thesis, supervised by Professor Michel Van Caneghem and defended on 14th June 2005 at University of Aix-Marseille II, France. The thesis, written in French, is available at http: //www.lif-sud.univ-mrs.fr/Rapports/25-2005.html. The mutual exclusion scheduling problem has an elegant graph-theoretic formulation: given an undirected graph G and an integer k, find a minimum coloring of G such that each color appears at most k times. When G is an interval graph, this problem has some applications in workforce planning. Then, the object of the thesis is to study the complexity of mutual exclusion scheduling problem for interval graphs and related classes. Received: August 2005 / Revised version: September 2005 Frédéric Gardi: On leave from Laboratoire d'Informatique Fondamentale - CNRS UMR 6166, Parc Scientifique et Technologique de Luminy, Marseille, France.     

Unbalanced bipartite factorizations of complete bipartite graphs

We construct a new infinite family of factorizations of complete bipartite graphs by factors all of whose components are copies of a (fixed) complete bipartite graph K_p,q. There are simple necessary conditions for such factorizations to exist. The family constructed here demonstrates sufficiency in many new cases. In particular, the conditions are always sufficient when q=p+1.