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### Labeling schemes for vertex connectivity

Amos Korman
2010 ACM Transactions on Algorithms
This paper studies labeling schemes for the vertex connectivity function on general graphs.  ...  The paper establishes an upper bound of k 2 log n on the number of bits used in a label. The best previous upper bound for the label size of such a labeling scheme is 2 k log n.  ...  In this section we present a k-vertex connectivity labeling scheme for the graph family G n using less than k 2 log n-bit labels.  ...

### Labeling Schemes for Vertex Connectivity [chapter]

Amos Korman
Lecture Notes in Computer Science
This paper studies labeling schemes for the vertex connectivity function on general graphs.  ...  The paper establishes an upper bound of k 2 log n on the number of bits used in a label. The best previous upper bound for the label size of such a labeling scheme is 2 k log n.  ...  In this section we present a k-vertex connectivity labeling scheme for the graph family G n using less than k 2 log n-bit labels.  ...

### Proximity-Preserving Labeling Schemes and Their Applications [chapter]

David Peleg
1999 Lecture Notes in Computer Science
This paper considers informative labeling schemes for graphs.  ...  It is shown that for the class of n-vertex weighted trees with M -bit edge weights, there exists such a proximity-preserving labeling scheme using O(M log n + log 2 n) bit labels.  ...  Acknowledgements I am grateful to Cyril Gavoille, Yehuda Hassin and Richard Tan for helpful discussions.  ...

### Fault-Tolerant Labeling and Compact Routing Schemes [article]

Michal Dory, Merav Parter
2021 arXiv   pre-print
Our main results for an n-vertex graph and a fault bound f are: – There is a randomized FT connectivity labeling scheme with a label length of O(f+log n) bits, hence optimal for f=O(log n).  ...  For a given n-vertex graph G and a bound f on the number of faults, an f-FT connectivity labeling scheme is a distributed data structure that assigns each of the graph edges and vertices a short label,  ...  For example, in routing applications the label of the vertex is treated as its "address". It is quite immediate to provide connectivity labeling schemes of logarithmic length.  ...

### A dynamic distributed approach to representing proper interval graphs

David Morgan
2009 Discrete Mathematics
scheme for proper interval graphs.  ...  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.  ...  Specifically, our labelling scheme is as follows. • For each vertex v, we maintain the following. -A unique identifier for each vertex.  ...

### Isometric universal graphs [article]

Louis Esperet, Cyril Gavoille, Carla Groenland
2021 arXiv   pre-print
Our main tool is a new type of distance labelling scheme, whose study might be of independent interest.  ...  We show that for any integer n≥ 1, there is a graph on 3^n+O(log^2 n) vertices that contains isometric copies of all n-vertex graphs.  ...  For any integer n 1, the class of all connected n-vertex graphs has a distance-vector labelling scheme with labels of at most 4n + O(log n) bits. Proof. Let G be a connected n-vertex graph.  ...

### Compact and localized distributed data structures

Cyril Gavoille, David Peleg
2003 Distributed computing
The survey concentrates mainly on combinatorial and algorithmic techniques, such as adjacency and distance labeling schemes and interval schemes for routing, and covers complexity results on various applications  ...  , focusing on compact localized schemes for message routing in communication networks.  ...  For edge-connectivity, this yields a tight bound of Θ(log 2 n) bits. Also, a k-vertex connectivity labeling scheme is given for general n-vertex graphs using O(log n) bit labels for fixed k.  ...

### Forbidden-Set Distance Labels for Graphs of Bounded Doubling Dimension

Ittai Abraham, Shiri Chechik, Cyril Gavoille, David Peleg
2016 ACM Transactions on Algorithms
For an n-vertex graph G in this family, and for any desired precision parameter > 0, the labeling scheme stores an O(1+ −1 ) 2α log 2 n-bit label at each vertex.  ...  The labeling scheme can be extended into a forbiddenset labeled routing scheme with stretch 1 + for graphs of bounded doubling dimension.  ...  Courcelle for helpful discussions about labeling schemes in the forbidden-set setting.  ...

### Forbidden-set distance labels for graphs of bounded doubling dimension

Ittai Abraham, Shiri Chechik, Cyril Gavoille, David Peleg
2010 Proceeding of the 29th ACM SIGACT-SIGOPS symposium on Principles of distributed computing - PODC '10
For an n-vertex graph G in this family, and for any desired precision parameter > 0, the labeling scheme stores an O(1+ −1 ) 2α log 2 n-bit label at each vertex.  ...  The labeling scheme can be extended into a forbiddenset labeled routing scheme with stretch 1 + for graphs of bounded doubling dimension.  ...  Courcelle for helpful discussions about labeling schemes in the forbidden-set setting.  ...

### Informative Labeling Schemes for Graphs [chapter]

David Peleg
2000 Lecture Notes in Computer Science
We begin by developing f labeling schemes for three functions f over the class of n-vertex trees.  ...  An f labeling scheme labels the vertices of a weighted graph G in such a way that f (W ) can be inferred (or at least approximated) efficiently for any vertex subset W of G by merely inspecting the labels  ...  Acknowledgements I am grateful to Michal Katz and Nir Katz for their helpful comments and suggestions.  ...

### Informative labeling schemes for graphs

David Peleg
2005 Theoretical Computer Science
We begin by developing f labeling schemes for three functions f over the class of n-vertex trees.  ...  An f labeling scheme labels the vertices of a weighted graph G in such a way that f (W ) can be inferred (or at least approximated) efficiently for any vertex subset W of G by merely inspecting the labels  ...  Acknowledgements I am grateful to Michal Katz and Nir Katz for their helpful comments and suggestions.  ...

### The vertex-transitive TLF-planar graphs [article]

D. Renault
2004 arXiv   pre-print
We characterize these graphs with a finite local representation and a special kind of finite state automaton named labeling scheme.  ...  We consider the class of the topologically locally finite (in short TLF) planar vertex-transitive graphs, a class containing in particular all the one-ended planar Cayley graphs and the normal transitive  ...  For the larger degrees, we enumerate all possible labeling schemes, each one corresponding to an enumerable family of planar vertex-transitive graphs.  ...

### Labeling Schemes with Queries [article]

Amos Korman, Shay Kutten
2006 arXiv   pre-print
We show that inspecting the label of a single additional vertex (one query) enables us to reduce the label size of many labeling schemes significantly.  ...  An f-labeling scheme for a family of graphs labels the vertices of all graphs in such that for every graph G∈ and every two vertices u,v∈ G, the value f(u,v) can be inferred by merely inspecting the labels  ...  They also show a labeling scheme supporting the k-vertex-connectivity function on general graphs using O(2 k log n)-bit labels. See [30] for a survey on (static) labeling schemes.  ...

### Enumerating planar locally finite Cayley graphs [article]

David Renault
2003 arXiv   pre-print
Keywords: vertex-transitive, Cayley graph, planar graph, tiling, labeling scheme  ...  We characterize the set of planar locally finite Cayley graphs, and give a finite representation of these graphs by a special kind of finite state automata called labeling schemes.  ...  The labeling scheme describes how the edges must be labeled : take a vertex s and an edge labeled by the generator a i pointing towards s.  ...

### Proof labeling schemes

Amos Korman, Shay Kutten, David Peleg
2005 Proceedings of the twenty-fourth annual ACM SIGACT-SIGOPS symposium on Principles of distributed computing - PODC '05
On the other hand, approaches are presented for the efficient construction of schemes, and upper and lower bounds are established on the cost of schemes for multiple basic problems.  ...  Our approach thus allows for a more modular design of algorithms, and has the potential to aid in verifying properties even when the original design of the structures for maintaining them was done without  ...  Note that the upper bound for k vertex connectivity is independent of k. Using similar methods we can construct proof labeling scheme for k-the flow between s and t, with size O(k log n).  ...
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