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20
The NPcompleteness column: an ongoing guide
 Journal of Algorithms
, 1985
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co ..."
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Cited by 188 (0 self)
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This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, crossreferences will be given to that book and the list of problems (NPcomplete and harder) presented there. Readers who have results they would like mentioned (NPhardness, PSPACEhardness, polynomialtimesolvability, etc.) or open problems they would like publicized, should
Distributed Loop Computer Networks: A Survey
, 1995
"... Distributed loop computer networks are extensions of the ring networks and are widely used in the design and implementation of local area networks and parallel processing architectures. We give a survey of recent results on this class of interconnection networks. We pay special attention to the actu ..."
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Cited by 74 (3 self)
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Distributed loop computer networks are extensions of the ring networks and are widely used in the design and implementation of local area networks and parallel processing architectures. We give a survey of recent results on this class of interconnection networks. We pay special attention to the actual computation of the minimum diameter and the construction of loop networks which can achieve this optimal number. Some open problems are offered for further investigation.
THE DIAMETER OF A CYCLE PLUS A RANDOM MATCHING
, 1988
"... How small can the diameter be made by adding a matching to an ncycle? In this paper this question is answered by showing that the graph consisting ofan ncycle and a random matching has diameter about log2n, which is very close to the best possible value. It is also shown that by adding a random ma ..."
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Cited by 67 (1 self)
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How small can the diameter be made by adding a matching to an ncycle? In this paper this question is answered by showing that the graph consisting ofan ncycle and a random matching has diameter about log2n, which is very close to the best possible value. It is also shown that by adding a random matching to graphs with certain expanding properties such as expanders or Ramanujan graphs, the resulting graphs have near optimum diameters.
Moore graphs and beyond: A survey of the degree/diameter problem
 ELECTRONIC JOURNAL OF COMBINATORICS
, 2013
"... The degree/diameter problem is to determine the largest graphs or digraphs of given maximum degree and given diameter. General upper bounds – called Moore bounds – for the order of such graphs and digraphs are attainable only for certain special graphs and digraphs. Finding better (tighter) upper bo ..."
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Cited by 26 (4 self)
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The degree/diameter problem is to determine the largest graphs or digraphs of given maximum degree and given diameter. General upper bounds – called Moore bounds – for the order of such graphs and digraphs are attainable only for certain special graphs and digraphs. Finding better (tighter) upper bounds for the maximum possible number of vertices, given the other two parameters, and thus attacking the degree/diameter problem ‘from above’, remains a largely unexplored area. Constructions producing large graphs and digraphs of given degree and diameter represent a way of attacking the degree/diameter problem ‘from below’. This survey aims to give an overview of the current stateoftheart of the degree/diameter problem. We focus mainly on the above two streams of research. However, we could not resist mentioning also results on various related problems. These include considering Moorelike bounds for special types of graphs and digraphs, such as vertextransitive, Cayley, planar, bipartite, and many others, on
A Note on Large Graphs of Diameter Two and Given Maximum Degree
"... Let vt(d; 2) be the largest order of a vertextransitive graph of degree d and diameter two. It is known that vt(d; 2) = d 2 + 1 for d = 1; 2; 3, and 7; for the remaining values of d we have vt(d; 2) d 2 \Gamma 1. The only known general lower bound on vt(d; 2), valid for all d, seems to be vt(d ..."
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Cited by 19 (5 self)
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Let vt(d; 2) be the largest order of a vertextransitive graph of degree d and diameter two. It is known that vt(d; 2) = d 2 + 1 for d = 1; 2; 3, and 7; for the remaining values of d we have vt(d; 2) d 2 \Gamma 1. The only known general lower bound on vt(d; 2), valid for all d, seems to be vt(d; 2) b d+2 2 cd d+2 2 e. Using voltage graphs, we construct a family of vertextransitive nonCayley graphs which shows that vt(d; 2) 8 9 (d + 1 2 ) 2 for all d of the form d = (3q \Gamma 1)=2 where q is a prime power congruent with 1 (mod 4). The construction generalizes to all prime powers and yields large highly symmetric graphs for other degrees as well. In particular, for d = 7 we obtain as a special case the HoffmanSingleton graph, and for d = 11 and d = 13 we have new largest graphs of diameter two and degree d on 98 and 162 vertices, respectively. 1 Introduction The wellknown degree/diameter problem asks for determining the largest possible number n(d; k) of vertic...
The Chromatic Number of Graph Powers
, 2000
"... Introduction The square G 2 of a graph G = (V; E) is the graph whose vertex set is V in which two distinct vertices are adjacent if and only if their distance in G is at most 2. What is the maximum possible chromatic number of G 2 , as G ranges over all graphs with maximum degree d and girth g? ..."
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Cited by 15 (0 self)
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Introduction The square G 2 of a graph G = (V; E) is the graph whose vertex set is V in which two distinct vertices are adjacent if and only if their distance in G is at most 2. What is the maximum possible chromatic number of G 2 , as G ranges over all graphs with maximum degree d and girth g? Our (somewhat surprising) answer is that for g = 3; 4; 5 or 6 this maximum is (1 + o(1))d 2 (where the o(1) term tends to 0 as d tends to infinity), whereas for all g 7, this maximum is of order d 2 = log d. To state this result more precisely, define, for every two integers d 2 and g<F9.9
Large Cayley Graphs and Digraphs with Small Degree and Diameter
, 1995
"... We review the status of the Degree#Diameter problem for both, graphs and digraphs and present new Cayley digraphs which yield improvements over some of the previously known largest vertex transitive digraphs of given degree and diameter. ..."
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Cited by 11 (0 self)
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We review the status of the Degree#Diameter problem for both, graphs and digraphs and present new Cayley digraphs which yield improvements over some of the previously known largest vertex transitive digraphs of given degree and diameter.
New large graphs with given degree and diameter
, 1992
"... Abstract: In this paper, a method for obtaining large diameter 6 graphs by replacing some vertices of a Moore bipartite diameter 6 graph with complete K h graphs is proposed. These complete graphs are joined to each other and to the remaining nonmodified graphs by means of new edges and by using a s ..."
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Cited by 7 (0 self)
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Abstract: In this paper, a method for obtaining large diameter 6 graphs by replacing some vertices of a Moore bipartite diameter 6 graph with complete K h graphs is proposed. These complete graphs are joined to each other and to the remaining nonmodified graphs by means of new edges and by using a special diameter 2 graph. The degree of the graph so constructed coincides with the original one. © 1999
Interconnection Topologies and Routing for Parallel Processing Systems
, 1992
"... The major aims of this work is to give a comparative survey of static interconnection topologies, and to discuss their properties with respect to their use as interconnection topologies in message passing multicomputer systems, i. e. each processing element has its own local memory, there is no com ..."
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Cited by 4 (0 self)
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The major aims of this work is to give a comparative survey of static interconnection topologies, and to discuss their properties with respect to their use as interconnection topologies in message passing multicomputer systems, i. e. each processing element has its own local memory, there is no common memory, and the processing elements communicate via messagepassing. To this end it was necessary to recall relevant measures on graphs from graph theory, like for example the average distance or the network diameter, and requirements from the parallel processing area, like the reliability or extensibility. Special emphasis has been given to present the construction rules for various graphs, because these seemed  along with the network characteristics  most relevant for interconnecting processing elements in reconfigurable multicomputer systems. Critical to applications in these kind of parallel systems is the possibility of exchanging local data among cooperating processing element...
A note on constructing large Cayley graphs of given degree and diameter by voltage assignments
, 1997
"... Voltage graphs are a powerful tool for constructing large graphs (called lifts) with prescribed properties as covering spaces of small base graphs. This makes them suitable for application to the degree/diameter problem, which is to determine the largest order of a graph with given degree and diame ..."
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Cited by 4 (1 self)
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Voltage graphs are a powerful tool for constructing large graphs (called lifts) with prescribed properties as covering spaces of small base graphs. This makes them suitable for application to the degree/diameter problem, which is to determine the largest order of a graph with given degree and diameter. Many currently known largest graphs of degree # 15 and diameter # 10 have been found by computer search among Cayley graphs of semidirect products of cyclic groups. We show that all of them can in fact be described as lifts of smaller Cayley graphs of cyclic groups, with voltages in (other) cyclic groups. # This research started when J. Plesnk and J. Siran were visiting the Department of Computer Science and Software Engineering of the University of Newcastle NSW Australia in 1995, supported by small ARC grant. the electronic journal of combinatorics 5 (1998), #R9 2 This opens up a new possible direction in the search for large vertextransitive graphs of given degree and diameter....