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### Ramsey numbers of graphs with long tails

S.A. Burr, Jerrold W. Grossman
1982 Discrete Mathematics
to an arbitrary n~~t~bipartit~ graph with a long path f'tail"i ~~lanat~ng from one of its points.  ...  While the classical ramsey numbers r(K,,, K,,1) seem beyond computation for all but very small values of n and m, much progress has been made [4] in determining ramsey numbers of sparser graphs, such  ...

### Explicit construction of linear sized tolerant networks

N. Alon, F.R.K. Chung
1988 Discrete Mathematics
graph still contains a path of length m.  ...  This settles a problem of Rosenberg, which was motivated by the study of fault tolerant linear arrays.  ...  Introduction What is the minimum possible number of vertices and edges of a graph G, such that even after removing all but portion of its vertices or edges, the remaining graph still contains a path of  ...

### Explicit construction of linear sized tolerant networks

N. Alon, F.R.K. Chung
2006 Discrete Mathematics
graph still contains a path of length m.  ...  This settles a problem of Rosenberg, which was motivated by the study of fault tolerant linear arrays.  ...  Introduction What is the minimum possible number of vertices and edges of a graph G, such that even after removing all but portion of its vertices or edges, the remaining graph still contains a path of  ...

### Explicit Construction of Linear Sized Tolerant Networks [chapter]

N. Alon, F.R.K. Chung
1988 Annals of Discrete Mathematics
graph still contains a path of length m.  ...  This settles a problem of Rosenberg, which was motivated by the study of fault tolerant linear arrays.  ...  Introduction What is the minimum possible number of vertices and edges of a graph G, such that even after removing all but portion of its vertices or edges, the remaining graph still contains a path of  ...

### On large induced trees and long induced paths in sparse random graphs

W.C.Stephen Suen
1992 Journal of combinatorial theory. Series B (Print)
Also, if L, denotes the length of the longest induced path in Gn,p, then we find a number h(c) such that L, is almost surely larger than (h(c) -E)n for any E>O.  ...  Let T" be the order of the largest induced subtree of G, p' We find a number t(c) such that T" is almost surely larger than (t(c) -&)n for any E > 0.  ...  INTRODUCTION AND MAIN RESULTS This paper is concerned with the number of vertices in the largest induced trees and the length of the longest induced paths in the well-known random graph G,, P.  ...

### Estimating graph distance and centrality on shared nothing architectures

Atilla Soner Balkir, Huseyin Oktay, Ian Foster
2014 Concurrency and Computation
Computing the exact shortest paths between a large number of vertices is a costly operation, and serial algorithms are not practical for billion-scale graphs.  ...  Our method exploits data locality, produces highly accurate results, and allows batch computation of shortest paths with 7% average error in graphs that contain billions of edges.  ...  The size of the induced graph is O.kD/, and the estimation algorithm runs in at most O.k 2 D 2 / time for a pair of vertices.  ...

### Page 4877 of Mathematical Reviews Vol. , Issue 89I [page]

1989 Mathematical Reviews
Specifically, the following characteristics are studied: the lengths of the longest cycle and the longest path, the total number of vertices in the unicyclic components and total number of vertices forming  ...  Moreover, the subgraph of a Ptolemy graph induced by the set of vertices having the maximal distance is a union of complete graphs.  ...

### Path Sampling: A Fast and Provable Method for Estimating 4-Vertex Subgraph Counts [article]

2014 arXiv   pre-print
Our algorithm is based on a novel technique of 3-path sampling and a special pruning scheme to decrease the variance in estimates.  ...  We provide theoretical proofs for the accuracy of our algorithm, and give formal bounds for the error and confidence of our estimates.  ...  So X is 1 iff the th call to sample outputs a 3-path contained in a copy of the ith motif. The total number of (distinct) such 3-paths is exactly A2,i · Ci.  ...

### Economic Genome Assembly from Low Coverage Illumina and Nanopore Data

Thomas Gatter, Sarah von Löhneysen, Polina Drozdova, Tom Hartmann, Peter F. Stadler, Nadia Pisanti, Carl Kingsford
2020 Workshop on Algorithms in Bioinformatics
In the next step, properties of proper interval graphs are used to extract contigs as maximum weight paths. These are translated into genomic sequences only in the final step.  ...  LazyB starts from a bipartite overlap graph between long reads and restrictively filtered short-read unitigs, which are then reduced to a long-read overlap graph G.  ...  In fact, F △ is an induced path in the triangle reduction H △ of H. This deduction suggests to identify maximal paths in the triangle reduction ⇀ G △ of the directed acycling graph ⇀ G as contigs.  ...

### Travel Planning with Self-Made Maps [chapter]

Ulrik Brandes, Frank Schulz, Dorothea Wagner, Thomas Willhalm
2001 Lecture Notes in Computer Science
To facilitate the use of such techniques when coordinates are missing from some, or even all, of the nodes in a network we generate artificial coordinates using methods from graph drawing.  ...  Experiments on a large set of German train timetables indicate that the speed-up achieved with coordinates from our network drawings is close to that achieved with the actual coordinates.  ...  For realistic data, the total number of springs thus introduced is linear in the number of vertices. Long-range dependencies.  ...

### Lightning Fast and Space Efficient k-clique Counting

Xiaowei Ye, Rong-Hua Li, Qiangqiang Dai, Hongzhi Chen, Guoren Wang
2022 Proceedings of the ACM Web Conference 2022
number of k-cliques in the dense regions of the graph.  ...  Existing exact k-clique counting algorithms are often hard to handle large dense graphs, while sampling-based solutions either require a huge number of samples or consume very high storage space to achieve  ...  Rong-Hua Li is the corresponding author of this paper.  ...

### SUMMA

Shijie Zhang, Shirong Li, Jiong Yang
2010 Proceedings of the 19th ACM international conference on Information and knowledge management - CIKM '10
The need for indexing massive graphs is an urgent research problem of great practical importance. The main challenge is the size. Each graph may contain at least tens of millions vertices.  ...  Graphs can represent a large number of data types, e.g., online social networks, internet links, procedure dependency graphs, etc.  ...  Also, we select those induced subgraphs of the database graph as query graphs.  ...

### A branch and bound algorithm for the robust shortest path problem with interval data

R. Montemanni, L.M. Gambardella, A.V. Donati
2004 Operations Research Letters
for each possible conÿguration of the arc costs.  ...  Many real problems can be modelled as robust shortest path problems on interval digraphs, where intervals represent uncertainty about real costs and a robust path is not too far from the shortest path  ...  They have very long paths from s to t (each path visits a number of vertices equal to the number of layers of the graph) and there is a huge number of alternative paths from s to t, since between two consecutive  ...

### Combinatorics and algorithms for augmenting graphs [article]

Konrad K. Dabrowski, Dominique de Werra, Vadim V. Lozin, Viktor Zamaraev
2014 arXiv   pre-print
However, our knowledge of augmenting graphs is still very limited, and we do not even know what the minimal infinite classes of augmenting graphs are.  ...  The notion of augmenting graphs generalizes Berge's idea of augmenting chains, which was used by Edmonds in his celebrated solution of the maximum matching problem.  ...  The support of EPFL is gratefully acknowledged.  ...

### Embedding nearly-spanning bounded degree trees

Noga Alon, Michael Krivelevich, Benny Sudakov
2007 Combinatorica
We derive a sufficient condition for a sparse graph G on n vertices to contain a copy of a tree T of maximum degree at most d on (1 − )n vertices, in terms of the expansion properties of G.  ...  We also prove that if an (n, D, λ)-graph G (i.e., a D-regular graph on n vertices all of whose eigenvalues, except the first one, are at most λ in their absolute values) has large enough spectral gap D  ...  They would like to thank the members of the Theory Group at Microsoft Research for their hospitality and for creating a stimulating research environment.  ...
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