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Coordinating Pebble Motion On Graphs, The Diameter Of Permutation Groups, And Applications

D. Kornhauser, G. Miller, P. Spirakis
1984 25th Annual Symposium onFoundations of Computer Science, 1984.  
We nlbo consider the rlucsf ion of permutation group diameter. Driscoli and I h r s t (1983) ohtairic4 a polynoinial upper bound on thc diameter of groups grnrrnt,rd by bounded lrrigth cyclrs.  ...  ,ounds on the numbei of Inovrs required. These resalts extend thosc of Wilson ;1974), who considered G biconnrctcd and k=n-I, with n o analysis of number of moves.  ...  The rest of chapter 3 consists of a number of new results in permutation groups, which extend classical theorems by providing upper bounds on diameter.  ... 
doi:10.1109/sfcs.1984.715921 dblp:conf/focs/KornhauserMS84 fatcat:qjqi77rq6bcsfhbfmx4nweezjq

Distance and connectivity measures in permutation graphs

Wayne Goddard, Michael E. Raines, Peter J. Slater
2003 Discrete Mathematics  
A permutation graph G π of a graph G (or generalized prism) is obtained by taking two disjoint copies of G and adding an arbitrary matching between the copies.  ...  In particular we show that if G has no isolates and is not 2K k for k odd, then there exists a permutation graph of G with edge-connectivity equal to its minimum degree.  ...  That is, F k is a permutation graph of the path on n = 2 k − 1 vertices, and has diameter about 4 log 2 n. 2 By a similar construction it holds that the minimum diameter of the permutation graph of the  ... 
doi:10.1016/s0012-365x(02)00870-1 fatcat:bnjnt5vvrvfa7hh33cimk66fza

Some problems on Cayley graphs

Elena Konstantinova
2008 Linear Algebra and its Applications  
This survey paper presents the historical development of some problems on Cayley graphs which are interesting to graph and group theorists such as Hamiltonicity or diameter problems, to computer scientists  ...  Acknowledgments I thank Reza Khosrovshahi for inviting me to participate at the First IPM (Institute for Studies in Theoretical Physics and Mathematics) conference on Algebraic Graph Theory that was held  ...  on April 21-26, 2007, in Tehran and for his hospitality during my staying in Iran.  ... 
doi:10.1016/j.laa.2008.05.010 fatcat:aw4jerefane77kfhanjxqeivvi

Computing the Diameter of 17-Pancake Graph Using a PC Cluster [chapter]

Shogo Asai, Yuusuke Kounoike, Yuji Shinano, Keiichi Kaneko
2006 Lecture Notes in Computer Science  
An n-pancake graph is a graph whose vertices are the permutations of n symbols and each pair of vertices are connected with an edge if and only if the corresponding permutations can be transitive by a  ...  Fundamental approaches of the diameter computation have been proposed. However, the computation of the diameter of 15-pancake graph has been the limit in practice.  ...  Tomoyuki Hiroyasu for permission to use the PC cluster on Doshisha University. A part of this research is supported by Japan society for the promotion of sciences, the grant-in-aid(No.16510105).  ... 
doi:10.1007/11823285_117 fatcat:gdhgbkqxlzf7hpntfdamshh7r4

Vertex reconstruction in Cayley graphs

Elena Konstantinova
2009 Discrete Mathematics  
The combinatorial properties of Cayley graphs on the symmetric group S n and the signed permutation group B n with respect to this problem are presented.  ...  In this report paper we collect recent results on the vertex reconstruction in Cayley graphs Cay(G, S).  ...  Let us note here that one more problem which will be discussed below arises on Cayley graphs. This is the problem of establishing the diameter of a Cayley graph.  ... 
doi:10.1016/j.disc.2008.07.039 fatcat:zbneleau4fauhgwgl7pwnicaem

On The Diameter of Pancake Graphs [article]

Harigovind V R, Pramod P Nair
2022 arXiv   pre-print
The diameter of a graph is the maximum of all distances on the graph, where the distance between two vertices is the shortest path between them.  ...  The Pancake graph(P_n) represents the group of all permutations on n elements, namely S_n, with respect to the generating set containing all prefix reversals.  ...  bound on the value of diam(Pn).  ... 
arXiv:2204.08184v1 fatcat:jiwgkfzj6rcfpelgst6tu7ff5y

A Comparative Study of Star Graphs and Rotator Graphs

Subburajan Ponnuswamy, Vipin Chaudhary
1994 1994 International Conference on Parallel Processing-Vol 1 (ICPP'94)  
Acknowledgements We would like to thank Ramaraghavan Srinivasan for his assistance with the simulation programs and for his comments on an earlier version of this paper.  ...  Since these directed Cayley graphs have some interesting properties like easy routing, low diameter and average diameter compared to the star graph, a comparative analysis of these graphs is worthy of  ...  The length of the container of any graph is the upper bound for the fault diameter of the graph 9, 10].  ... 
doi:10.1109/icpp.1994.18 dblp:conf/icpp/PonnuswamyC94 fatcat:hktb4l5yr5ggpangfpfimo5bri

A 1.375-Approximation Algorithm for Sorting by Transpositions

Isaac Elias, Tzvika Hartman
2006 IEEE/ACM Transactions on Computational Biology & Bioinformatics  
The algorithm is based on a new upper bound on the diameter of 3-permutations.  ...  of 2-permutations and simple permutations.  ...  We prove an upper bound of 11 n 24 + 3 (n/3 mod 8) 2 + 1 on the diameter of 3-permutations.  ... 
doi:10.1109/tcbb.2006.44 pmid:17085846 fatcat:riw53h4j6zeonhcivqhca6j6qu

A 1.375-Approximation Algorithm for Sorting by Transpositions [chapter]

Isaac Elias, Tzvika Hartman
2005 Lecture Notes in Computer Science  
The algorithm is based on a new upper bound on the diameter of 3-permutations.  ...  of 2-permutations and simple permutations.  ...  We prove an upper bound of 11 n 24 + 3 (n/3 mod 8) 2 + 1 on the diameter of 3-permutations.  ... 
doi:10.1007/11557067_17 fatcat:thxu6wgs2fdkta33wczo5y4pmq

Oriented Diameter of Star Graphs [article]

K. S. Ajish Kumar, Deepak Rajendraprasad, K. S. Sudeep
2019 arXiv   pre-print
The n-star graph S_n consists of n! vertices, each labelled with a distinct permutation of [n]. Two vertices are adjacent if their labels differ exactly in the first and one other position.  ...  An orientation of an undirected graph G is an assignment of exactly one direction to each edge of G.  ...  [9] continues the algorithmic study on oriented diameter on chordal graphs. They show that every chordal graph G has an oriented diameter at most 2 diam(G) + 1.  ... 
arXiv:1911.10340v1 fatcat:xq6gmpjfbnazlfwvcy43jzcyoa

On the diameter of Eulerian orientations of graphs

László Babai
2006 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm - SODA '06  
The result also shows that the directed and undirected versions of long standing conjectures regarding the diameter of Cayley graphs of various classes of groups, including transitive permutation groups  ...  It follows that recent polynomial bounds on the diameter of certain large classes of Cayley graphs of the symmetric group and certain linear groups automatically extend to directed Cayley graphs.  ...  Distance in and diameter of Cayley graphs of permutation groups are computationally hard parameters.  ... 
doi:10.1145/1109557.1109648 fatcat:gna44cf5tbabnp7rlxt44dcg3e

Processor interconnection networks from Cayley graphs

Stephen T. Schibell, Richard M. Stafford
1992 Discrete Applied Mathematics  
The importance of the average diameter in determining network performance was supported by the fact that the graphs found by us had the smallest average diameter and outperformed all other graphs evaluated  ...  to produce a fast efficient routing algorithm on the associated Cayley graph.  ...  Robert Morris for his encouragement and assistance through the research phase of this work as well as during the preparation of this paper.  ... 
doi:10.1016/0166-218x(92)90005-u fatcat:xe5xavk6jbhg5hw77hrcst7czq

Diameter of the commutation classes graph of a permutation [article]

G. Gutierres, R. Mamede, J.L. Santos
2022 arXiv   pre-print
We define a statistic on the graph of commutation classes of a permutation of the symmetric group which is used to show that these graphs are equipped with a ranked poset structure, with a minimum and  ...  This characterization also allows us to compute the diameter of the commutation graph for any permutation, from which the results for the longest permutation and for fully commutative permutations are  ...  one) of the permutation in the line above.  ... 
arXiv:2202.07313v1 fatcat:y43pruvpq5fwtbqnsjimtnu2zq

Computing short generator sequences

James R. Driscoll, Merrick L. Furst
1987 Information and Computation  
The Cayley graph is the graph whose vertices are the elements of G, with edges between vertices p and q iff pg, = q, for some generator gj. The diameter of a group is very dependent on presentation.  ...  Another way to say the same thing is to say that diam(G) is the graph-theoretic diameter of the Cayley graph f(G).  ... 
doi:10.1016/0890-5401(87)90043-5 fatcat:myrqklx3unbt3l4oh3c6akqnqm

Bounds for the diameters of orbital graphs of affine groups [article]

Attila Maróti, Saveliy V. Skresanov
2022 arXiv   pre-print
General bounds are presented for the diameters of orbital graphs of finite affine primitive permutation groups.  ...  Several infinite families of affine primitive permutation groups with large orbital diameter are constructed. The results are independent from the classification of finite simple groups.  ...  Acknowledgement The authors would like to thank Sean Eberhard and László Pyber for comments on an earlier version of the paper.  ... 
arXiv:2205.03593v1 fatcat:zh65tgqt3rdjxguxumbyqojljm
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