## Implicit Representation of Graphs (1992)

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Venue: | SIAM Journal On Discrete Mathematics |

Citations: | 71 - 0 self |

### BibTeX

@INPROCEEDINGS{Kannan92implicitrepresentation,

author = {Sampath Kannan and Moni Naor and Steven Rudich},

title = {Implicit Representation of Graphs},

booktitle = {SIAM Journal On Discrete Mathematics},

year = {1992},

pages = {334--343}

}

### Years of Citing Articles

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### Abstract

How to represent a graph in memory is a fundamental data structuring question. In the usual representations of an n-vertex graph, the names of the vertices (i.e. integers from 1 to n) betray nothing about the graph itself. Indeed, the names (or labels) on the n vertices are just log n bit place holders to allow data on the edges to encode the structure of the graph. In our scenario, there is no such waste. By assigning O(log n) bit labels to the vertices, we completely encode the structure of the graph, so that given the labels of two vertices we can test if they are adjacent in time linear in the size of the labels. Furthermore, given an arbitrary original labeling of the vertices, we can find structure coding labels (as above) that are no more than a small constant factor larger than the original labels. These notions are intimately related to vertex induced universal graphs of polynomial size. For example, we can label planar graphs with structure coding labels of size ! 4 log n, which implies the existence of a graph with n 4 vertices that contains all n-vertex planar graphs as vertex induced subgraphs.

### Citations

1134 |
Algorithmic Graph Theory and Perfect Graphs
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(Show Context)
Citation Context ... us a labeling scheme. This is so for a number of families: circle graphs, circular arc graphs, permutation graphs, graphs with bounded interval number. See Golumbic for definitions of these families =-=[8]-=-. Path Graphs: Path graphs are graphs where each vertex is represented by a path in a tree. Two vertices are adjacent iff the paths representing them intersect. There are several kinds of path graphs ... |

134 |
A tradeoff between space and efficiency for routing tables
- Peleg, Upfal
- 1988
(Show Context)
Citation Context ...thermore from the representation of any graph it is easy to derive the representation of any of its vertex induced subgraphs. Fredrickson and Janardan [6], Santoro and Khatib [14] and Peleg and Upfal =-=[12]-=- considered the question of storing routing information at the vertices of a packet switching network so as to compute near-optimal routes. Again, they look at the overall storage requirements 9 rathe... |

134 |
Labelling and Implicit Routing in Networks
- Santoro, Khatib
- 1985
(Show Context)
Citation Context ...o labelable families. Furthermore from the representation of any graph it is easy to derive the representation of any of its vertex induced subgraphs. Fredrickson and Janardan [6], Santoro and Khatib =-=[14]-=- and Peleg and Upfal [12] considered the question of storing routing information at the vertices of a packet switching network so as to compute near-optimal routes. Again, they look at the overall sto... |

71 | Compact distributed data structures for adaptive routing
- AWERBUCH, BAR-NOY, et al.
- 1989
(Show Context)
Citation Context ...utes. Again, they look at the overall storage requirements 9 rather than the requirements on a per vertex basis. This work has recently been extended to limit the amount of storage in every processor =-=[1]-=-. A number of papers have considered the question of universal graphs for a family of graphs. Most of these papers have focused on edge induced universal graphs which have often just been called unive... |

61 |
Edge-disjoint spanning trees of finite graphs
- Nash-Williams
- 1961
(Show Context)
Citation Context ...ertex induced subgraphs of G. Then the arboricity of G is defined to be max H jE(H)j jV (H)j \Gamma 1 where jE(H)j is the number of edges in H and jV (H)j is the number of vertices in H. NashWilliams =-=[11]-=- showed that the edges of a graph G with arboricity k can be decomposed into k forests. This means that there is a k + 1 labeling scheme for graphs of arboricity k. Prelabel the vertices arbitrarily w... |

25 |
An unexpected result on coding the vertices of a graph
- Breuer, Folkman
- 1967
(Show Context)
Citation Context ...on-trivial example, planar graphs have a 4-labeling scheme and hence have n 4 sized universal graphs. 6 Related Work Several authors have worked on related problems. Breuer [3] and Breuer and Folkman =-=[4]-=- considered the problem of labeling vertices such that adjacency would be determined by the Hamming distance of the labels. This is a restricted labeling scheme. They prove that this can be done for a... |

23 |
Coding vertexes of a graph
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- 1966
(Show Context)
Citation Context ...about the embedding. As a non-trivial example, planar graphs have a 4-labeling scheme and hence have n 4 sized universal graphs. 6 Related Work Several authors have worked on related problems. Breuer =-=[3]-=- and Breuer and Folkman [4] considered the problem of labeling vertices such that adjacency would be determined by the Hamming distance of the labels. This is a restricted labeling scheme. They prove ... |

22 | Universal graphs for bounded-degree trees and planar graphs
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- 1989
(Show Context)
Citation Context ... edge induced universal for a set of graphs S if every graph in S is a subgraph of G. The important issue in constructing edge induced universal graphs is minimizing the number of edges. Bhatt et al. =-=[2]-=- consider the question of constructing n vertex edge induced universal graphs for n vertex bounded degree trees and planar graphs. They show that for bounded degree trees there is an edge induced univ... |

19 |
Separator-based strategies for efficient message routing
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- 1986
(Show Context)
Citation Context ...G is c-decomposable if for all subgraphs H with more than c vertices, there exist c vertices such that their removal causes H to be disconnected with no component containing more than 2jHj=3 vertices =-=[6]-=-. To label a c-decomposable graph G we construct a tree decomposition T, of G in the following manner. Let S be a c-separator for G. Then S is assigned to the root of T. If H 1 and H 2 are the two com... |

16 |
Intersection Graphs of Paths in a Tree
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- 1986
(Show Context)
Citation Context ...represented by a path in a tree. Two vertices are adjacent iff the paths representing them intersect. There are several kinds of path graphs mentioned in the literature. See Monma and Wei for details =-=[10]-=-. We will 5 show how to label path graphs where the paths are undirected and two paths intersect if they have a vertex in common. Similar methods will work for all path graph families, provided the pa... |

5 |
A tradeo between space and e ciency for routing tables
- Peleg, Upfal
- 1989
(Show Context)
Citation Context ...thermore from the representation of any graph it is easy to derive the representation of any of its vertex induced subgraphs. Fredrickson and Janardan [6], Santoro and Khatib [14] and Peleg and Upfal =-=[12]-=- considered the question of storing routing information at the vertices of a packet switching network so as to compute near-optimal routes. Again, they look at the overall storage requirements 9rathe... |

2 |
Edge-disjoint spanning trees of nite graphs
- Nash-Williams
- 1961
(Show Context)
Citation Context ...ible vertex induced subgraphs of G. Then the arboricity of G is de ned to be max H jE(H)j jV (H)j � 1 where jE(H)j is the number of edges in H and jV (H)j is the number of vertices in H. NashWilliams =-=[11]-=- showed that the edges of a graph G with arboricity k can be decomposed into k forests. This means that there is a k + 1 labeling scheme for graphs of arboricity k. Prelabel the vertices arbitrarily w... |

1 |
Universal graphs and induced universal graphs, preprint
- Chung
(Show Context)
Citation Context ...niversal graph with cn edges where c is a constant depending only on the degree bound. For bounded degree planar graphs they show an edge induced universal graph with O(n log n) edges. Recently Chung =-=[5]-=- has shown how to construct an edge induced universal graph for n vertex trees having n vertices and O(n log n) edges. For planar graphs [5] gives a bound of O(n 3 =2) on the number of edges in the ed... |

1 |
A recognition algorithm for the intersection of paths in trees
- Gavril
- 1978
(Show Context)
Citation Context ... an end vertex of the other path. This requires at most six adjacency tests in a transitive closure of a tree. Finding the path representation and thus the labeling of a graph can be done efficiently =-=[7]-=-. It is interesting to note that the closely related family of chordal graphs which can be characterized as the intersection of subtrees in a tree cannot be labeled. 3.4 c-Decomposable Graphs Definiti... |

1 |
Succinct representation of graphs, Disc
- Turan
- 1984
(Show Context)
Citation Context ...he Hamming distance of the labels. This is a restricted labeling scheme. They prove that this can be done for any graph; however, the length of the labels could be very large compared to log n. Turan =-=[15]-=- considered the problem of representing a graph as succinctly as possible. But the representation is global; it gives an efficient representation for the whole graph without necessarily partitioning t... |

1 |
Separator-based strategies for e cient message routing
- Fredrickson, Janardan
- 1986
(Show Context)
Citation Context ...G is c-decomposable if for all subgraphs H with more than c vertices, there exist c vertices such that their removal causes H to be disconnected with no component containing more than 2jHj=3 vertices =-=[6]-=-. To label a c-decomposable graph G we construct a tree decomposition T, of G in the following manner. Let S be a c-separator for G. Then S is assigned to the root of T. If H1 and H2 are the two compo... |