A sufficient condition for pancyclic graphs

In 2005, Rahman and Kaykobad proved that if G is a connected graph of order n such that d(x)+d(y)+d(x,y)n+1 for each pair x, y of distinct nonadjacent vertices in G, where d(x,y) is the length of a shortest path between x and y in G, then G has a Hamiltonian path [Inform. Process. Lett. 94 (2005) 37–41]. In 2006 Li proved that if G is a 2-connected graph of order n3 such that d(x)+d(y)+d(x,y)n+2 for each pair x,y of nonadjacent vertices in G, then G is pancyclic or G=K_n/2,n/2 where n4 is an even integer [Inform. Process. Lett. 98 (2006) 159–161]. In this work we prove that if G is a 2-connected graph of order n such that d(x)+d(y)+d(x,y)n+1 for all pairs x, y of distinct nonadjacent vertices in G, then G is pancyclic or G belongs to one of four specified families of graphs.     

Note on the connectivity of line graphs

Let G be a connected graph with vertex set V(G), edge set E(G), vertex-connectivity κ(G) and edge-connectivity λ(G).A subset S of E(G) is called a restricted edge-cut if G−S is disconnected and each component contains at least two vertices. The restricted edge-connectivity λ₂(G) is the minimum cardinality over all restricted edge-cuts. Clearly λ₂(G)?λ(G)?κ(G).In 1969, Chartrand and Stewart have shown that

     

带权区间图的最短路算法

提出一个解带权区间图的最短路问题的O(nα(n))时间新算法，其中n是带权区间图中带权区间的个数，α(n)是单变量Ackerman函数的逆函数，它是一个增长速度比log n慢得多的函数，对于通常所见到的n，α(n)≤4．本文提出的新算法不仅在时间复杂性上比直接用Dijkstra算法解带权区间图的最短路问题有较大改进，而且算法设计思想简单，易于理解和实现．     

Shortest noncrossing paths in plane graphs

Let G be an undirected plane graph with nonnegative edge length, and letk terminal pairs lie on two specified face boundaries. This paper presents an algorithm for findingk noncrossing paths inG, each connecting a terminal pair, and whose total length is minimum. Noncrossing paths may share common vertices or edges but do not cross each other in the plane. The algorithm runs in timeO(n logn) wheren is the number of vertices inG andk is an arbitrary integer.     

Output-Sensitive Reporting of Disjoint Paths

A k -path query on a graph consists of computing k vertex-disjoint paths between two given vertices of the graph, whenever they exist. In this paper we study the problem of performing k -path queries, with , in a graph G with n vertices. We denote with the total length of the reported paths. For , we present an optimal data structure for G that uses O(n) space and executes k -path queries in output-sensitive time. For triconnected planar graphs, our results make use of a new combinatorial structure that plays the same role as bipolar (st ) orientations for biconnected planar graphs. This combinatorial structure also yields an alternative construction of convex grid drawings of triconnected planar graphs. Received August 24, 1996; revised April 8, 1997.     

A new sufficient condition for Hamiltonicity of graphs

Rahman and Kaykobad proved the following theorem on Hamiltonian paths in graphs. Let G be a connected graph with n vertices. If d(u)+d(v)+δ(u,v)?n+1 for each pair of distinct non-adjacent vertices u and v in G, where δ(u,v) is the length of a shortest path between u and v in G, then G has a Hamiltonian path. It is shown that except for two families of graphs a graph is Hamiltonian if it satisfies the condition in Rahman and Kaykobad's theorem. The result obtained in this note is also an answer for a question posed by Rahman and Kaykobad.     

An optimal algorithm for shortest paths on weighted interval and circular-arc graphs,with applications

We give the first linear-time algorithm for computing single-source shortest paths in a weighted interval or circular-arc graph, when we are given the model of that graph, i.e., the actual weighted intervals or circular-arcsand the sorted list of the interval endpoints. Our algorithm solves this problem optimally inO(n) time, wheren is the number of intervals or circular-arcs in a graph. An immediate consequence of our result is anO(qn + n logn)-time algorithm for the minimum-weight circle-cover problem, whereq is the minimum number of arcs crossing any point on the circle; then logn term in this time complexity is from a preprocessing sorting step when the sorted list of endpoints is not given as part of the input. The previously best time bounds were0(n logn) for this shortest paths problem, andO(qn logn) for the minimum-weight circle-cover problem. Thus we improve the bounds of both problems. More importantly, the techniques we give hold the promise of achieving similar (logn)-factor improvements in other problems on such graphs.The research of M. J. Atallah was supported in part by the Leonardo Fibonacci Institute, Trento, Italy, by the Air Force Office of Scientific Research under Contract AFOSR-90-0107, and by the National Science Foundation under Grant CCR-9202807. D. Z. Chen's research was supported in part by the Leonardo Fibonacci Institute, Trento, Italy. The research of D. T. Lee was supported in part by the Leonardo Fibonacci Institute, Trento, Italy, by the National Science Foundation, and the Office of Naval Research under Grants CCR-8901815, CCR-9309743, and N00014-93-1-0272.     

Embedding k(n−k) edge-disjoint spanning trees in arrangement graphs

The arrangement graphs are a class of generalized star graphs. In this paper we construct a graph that consists of the maximum number of directed edge-disjoint spanning trees in an arrangement graph. The paths that connect the common root node to any given node through different spanning trees are node-disjoint, and the lengths of these paths differ from the shortest possible lengths by a small additive constant. This graph can be used to derive fault-tolerant algorithms for broadcasting and scattering problems without prior knowledge of the faulty elements of the network.     

Function-described graphs for modelling objects represented by sets of attributed graphs

We present in this article the model function-described graph (FDG), which is a type of compact representation of a set of attributed graphs (AGs) that borrow from random graphs the capability of probabilistic modelling of structural and attribute information. We define the FDGs, their features and two distance measures between AGs (unclassified patterns) and FDGs (models or classes) and we also explain an efficient matching algorithm. Two applications of FDGs are presented: in the former, FDGs are used for modelling and matching 3D-objects described by multiple views, whereas in the latter, they are used for representing and recognising human faces, described also by several views.     

On Hamiltonian Paths and a Chain Metric

A recurrent formula and an ordinary formula are proposed for the number of Hamiltonian paths on special digraphs generated by a chain metric on n-permutations.     

Sufficient conditions for connectivity maintenance and rendezvous in leader-follower networks

In this paper we derive a set of constraints that are sufficient to guarantee maintained connectivity in a leader-follower multi-agent network with a proximity based communication topology. In the scenario we consider, only the leaders are aware of the global mission, which is to converge at a known destination point. Thus, the followers need to stay in contact with the group of leaders in order to reach the goal. In the paper we show that we can maintain the initial network structure, and thereby connectivity, by setting up bounds on the ratio of leaders-to-followers and on the magnitude of the goal attraction force experienced by the leaders. The results are first established for an initially complete communication graph and then extended to an incomplete graph. The results are illustrated by computer simulations.     

巧用Photoshop路径制作花式图案

在简要介绍了Photoshop中的路径工具后,利用其钢笔工具与形状路径详细介绍了花式图案的制作过程,并对制作过程中的巧妙用法进行了归纳.     

Supereulerian graphs with small matching number and 2-connected hamiltonian claw-free graphs

Motivated by the Chinese Postman Problem, Boesch, Suffel, and Tindell [The spanning subgraphs of Eulerian graphs, J. Graph Theory 1 (1977), pp. 79–84] proposed the supereulerian graph problem which seeks the characterization of graphs with a spanning Eulerian subgraph. Pulleyblank [A note on graphs spanned by Eulerian graphs, J. Graph Theory 3 (1979), pp. 309–310] showed that the supereulerian problem, even within planar graphs, is NP-complete. In this paper, we settle an open problem raised by An and Xiong on characterization of supereulerian graphs with small matching numbers. A well-known theorem by Chvátal and Erdös [A note on Hamilton circuits, Discrete Math. 2 (1972), pp. 111–135] states that if G satisfies α(G)≤κ(G), then G is hamiltonian. Flandrin and Li in 1989 showed that every 3-connected claw-free graph G with α(G)≤2 κ(G) is hamiltonian. Our characterization is also applied to show that every 2-connected claw-free graph G with α(G)≤3 is hamiltonian, with only one well-characterized exceptional class.     

Faster recognition of clique-Helly and hereditary clique-Helly graphs

A family of subsets of a set is Helly when every subfamily of it, which is formed by pairwise intersecting subsets contains a common element. A graph G is clique-Helly when the family of its (maximal) cliques is Helly, while G is hereditary clique-Helly when every induced subgraph of it is clique-Helly. The best algorithms currently known to recognize clique-Helly and hereditary clique-Helly graphs have complexities O(nm²) and O(n²m), respectively, for a graph with n vertices and m edges. In this Note, we describe algorithms which recognize both classes in O(m²) time. These algorithms also reduce the complexity of recognizing some other classes, as disk-Helly graphs.     

MANET中QoS路径可靠性的分析构架

文章研究了MANET中多径特性对QoS路径可靠性的影响,针对多跳路径的可靠性计算问题,提出了一种分析构架。几个考虑的主要参数有平均链路簇,多径数目,路径复杂度和跳数差别因子。方案具有一定的普遍适用性,在给定的源和目的节点中,对于单路径和多径的情况都能较好运用。     

Different Algorithms for Normal and Protection Paths

Many network routing situations commonly require backup paths that satisfy various constraints on bandwidth, link or node selection, and ease of configuration. In this paper, we attempt to validate whether it is beneficial to have distinct algorithmic treatments of normal and backup path calculation, configuration, and maintenance. We present a modular suite of algorithms that enable us to manage normal and protection paths differently. In the process, we develop a simple extension of Minimum Interference Routing Algorithm for shared protection paths. We incorporate a distributed algorithm to separately calculate normal and backup paths in the network, using link state information, and present an evaluation of asynchronous dynamic reorganization of backup paths to reduce congestion in the network. Simulations demonstrate nontrivial quantitative reductions in blocking probabilities under certain conditions. We conclude that in order to choose an optimal algorithm for a protected QoS routing application, it is recommended to also consider a combination of two different algorithms for normal and backup paths.Rajarshi Gupta is a PhD candidate in Electrical Engineering and Computer Sciences at the University of California at Berkeley and will graduate in May 2005. Prior to this, he completed hisMS degree in 1999 at Berkeley and his BS degree in 1997 from the University of Maryland. From 1999 to 2003, Rajarshi worked with Extreme Networks as a Senior Designer, where he has been the author of 8 patents. He is interested in algorithms to ensure quality in networks–both wired and ad-hoc. This includes: analysis of network capacity; switching and scheduling mechanisms for efficient utilization of resources; and, routing algorithms to guarantee quality of service.Eric Chi received his MS degree in Electrical Engineering and Computer Sciences from the University of California at Berkeley in 2001 and his BAdegree in physics in 1999 from Rice University. Hiswork focused on distributed network capacity management.He hasworked on inventory restocking problems and protection path resource allocation in wired communication networks.Jean Walrand received the PhD degree from the Department of Electrical Engineering and Computer Sciences of the University of California at Berkeley where he is now Professor. His research interests include decision theory, stochastic processes, and communication networks. He is the author of An Introduction to Queueing Networks (Prentice Hall, 1988) and of Communication Networks: A First Course (2nd ed. McGraw-Hill, 1998) and coauthor of High-Performance Communication Networks (2nd ed, Morgan Kaufman, 2000). He is a Fellow of the Belgian American Education Foundation and of the IEEE and a recipient of the Lanchester Prize and of the Stephen O. Rice Prize.     

无线网络中寻找非干扰不相交路径的拟人算法

针对无线网络中寻找从源点s到汇点t的两条非干扰不相交路径这一NP难问题,提出了一种拟人算法。该算法首先基于网络流方法得到两条点不相交的s-t路径,然后通过一种拟人化的策略逐步调整这两条路径,力图使得它们变为两条非干扰不相交的s-t路径。模拟实验表明,与现有的算法相比,拟人算法可以快速地以更高的概率找到两条长度较短的非干扰不相交路径。     

Spanners in sparse graphs

A t-spanner of a graph G is a spanning subgraph S in which the distance between every pair of vertices is at most t times their distance in G. If S is required to be a tree then S is called a tree t-spanner of G. In 1998, Fekete and Kremer showed that on unweighted planar graphs deciding whether G admits a tree t-spanner is polynomial time solvable for t?3 and is NP-complete when t is part of the input. They also left as an open problem if the problem is polynomial time solvable for every fixed t?4. In this work we resolve the open question of Fekete and Kremer by proving much more general results:

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  The problem of finding a t-spanner of treewidth at most k in a given planar graph G is fixed parameter tractable parameterized by k and t. Moreover, for every fixed t and k, the running time of our algorithm is linear.

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  Our technique allows to extend the result from planar graphs to much more general classes of graphs. An apex graph is a graph that can be made planar by the removal of a single vertex. We prove that the problem of finding a t-spanner of treewidth k is fixed parameter tractable on graphs that do not contain some fixed apex graph as a minor, i.e. on apex-minor-free graphs. The class of apex-minor-free graphs contains planar graphs and graphs of bounded genus.

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  Finally, we show that the tractability border of the t-spanner problem cannot be extended beyond the class of apex-minor-free graphs and in this sense our results are tight. In particular, for every t?4, the problem of finding a tree t-spanner is NP-complete on K6-minor-free graphs.