## A Structural Approach to Graph Compression (1998)

Venue: | In MFCS Workshop on Communications |

Citations: | 14 - 1 self |

### BibTeX

@INPROCEEDINGS{Deo98astructural,

author = {Narsingh Deo and Bruce Litow},

title = {A Structural Approach to Graph Compression},

booktitle = {In MFCS Workshop on Communications},

year = {1998},

pages = {91--101}

}

### Years of Citing Articles

### OpenURL

### Abstract

We consider graph compression in terms of graph families. In particular, we show that graphs of bounded genus can be compressed to O(n) bits, where n is the number of vertices. We identify a property based on separators that makes O(n)-bit compression possible for some graphs of bounded arboricity. 1 Introduction Graph representation as a data compression problem Lossless data compression is a process of representing a body of data by another body of data of smaller size from which the original data can be completely reconstructed. In the past thirty years a great deal of work has been done on the theory and practice of text compression (e.g., printed text or program source code) and of digitized data (e.g., voice or images). In fact, data compression has become a well-established subject in computer science, information theory, and communication theory. In contrast, very little has been done on compressing graphs. Since graphs are encountered everywhere and are often of very la...

### Citations

2438 |
The Design and Analysis of Computer Algorithms
- Aho, Hopcroft, et al.
- 1974
(Show Context)
Citation Context ...ere are exactly 2 6 = 64 labeled graphs; 11 nonisomorphic unlabeled graphs; but note that the lower bound is d 64 4! e = 3. The standard adjacency list representation for G requires (n+2m) log n bits =-=[1]-=-. When m ? 3n 2 8 log n one can do better than this, as follows [16]: partition V into V 1 and V 2 of size n=2 each, and let G 1 and G 2 be the vertex-induced subgraphs on V 1 and V 2 , respectively. ... |

1789 | Random Graphs
- Bollob'as
- 1981
(Show Context)
Citation Context ... argument similar to the one for all graphs shows that log NF (n) = \Theta(n log n) : In this case, C f (n) = O(1). The subset of F consisting of c-regular graphs still contains roughly n cn=2 graphs =-=[5]-=-, so that over this subset C f (n) = O(1). These examples suggest that the adjacency list representation is essentially optimal for sparse graphs. In the next two sections we will provide evidence tha... |

389 | A Separator Theorem for Planar Graphs
- Lipton, Tarjan
- 1979
(Show Context)
Citation Context ... separator if removal of C results in a disconnected graph. Let us recall some separator theorems for some of the important graph families. The first result applies to planar graphs and was proved in =-=[13]-=-. Theorem 4 If G is a planar graph, then there is a separator C such that the vertex set of G can be partitioned as A [ B [ C where jCjsfin 1=2 , (fi is a constant), jAj; jBjs2n=3, and no edge of G co... |

236 |
A Group Theoretic Model for Symmetric Interconnection Networks
- Akers, Krishnamurthy
- 1989
(Show Context)
Citation Context ...morphism group. The family of vertex transitive graphs is a natural example of this kind of structure. Its subfamily of Cayley graphs, and special types of Cayley graphs have been widely studied. See =-=[2, 3, 14]-=- for further examples and references. Graphs whose interconnection patterns are known in advance have been constructed and studied, especially as the basis of parallel and distributed computing archit... |

153 |
The star graph: an attractive alternative to the n-cube
- Akers, Harel, et al.
- 1987
(Show Context)
Citation Context ...morphism group. The family of vertex transitive graphs is a natural example of this kind of structure. Its subfamily of Cayley graphs, and special types of Cayley graphs have been widely studied. See =-=[2, 3, 14]-=- for further examples and references. Graphs whose interconnection patterns are known in advance have been constructed and studied, especially as the basis of parallel and distributed computing archit... |

95 | A separator theorem for graphs with an excluded minor and its applications
- Alon, Seymour, et al.
- 1990
(Show Context)
Citation Context ...g, then a partition A; B; C of the vertices of G can be computed in O(m) time such that jAj; jBjs2n=3, jCj = O(gn) 1=2 , and no edge of G connects a vertex of A to a vertex of B. Next, we recall from =-=[4]-=- a separator theorem for graphs defined by an excluded minor. A graph H is said to be a minor of a graph G iff H can be obtained from G by (possibly) removing some vertices and edges, and (possibly) c... |

81 |
A separator theorem for graphs of bounded genus
- Gilbert, Hutchinson, et al.
- 1984
(Show Context)
Citation Context ...A [ B [ C where jCjsfin 1=2 , (fi is a constant), jAj; jBjs2n=3, and no edge of G connects a vertex of A to a vertex of B. The separator C can be computed in O(n) time. Theorem 4 has been extended in =-=[8]-=- to graphs of genus g in the following way. Theorem 5 If G is a graph of genus g, and G is imbedded in a surface of genus g, then a partition A; B; C of the vertices of G can be computed in O(m) time ... |

57 |
The graph genus problem is NP-complete
- Thomassen
- 1989
(Show Context)
Citation Context ...that the problem of finding an imbedding of G in a surface of minimal genus is intractable under the assumption that P 6= NP since Thomassen has shown that finding the genus of a graph is NP-complete =-=[17]-=-. We can use book embedding as the basis for a compact, efficiently computable representation for graphs of bounded genus. Theorem 3 Given an imbedding of a graph G of genus g in a surface, a represen... |

34 |
On well-partial-order theory and its applications to combinatorial problems of VLSI design
- Fellows, Langston
- 1992
(Show Context)
Citation Context ...size at most n=2. The separator can be computed in O(h 1=2 n 1=2 (n +m)) time. 6 We conclude this subsection with a theorem relating edge density and excluded minors. It is essentially Theorem 2.5 of =-=[7]-=-. Theorem 7 Let F be the family of graphs excluding a given graph H as a minor. There is a constant !H such that every member of F has arboricity at most !H . Proof : Theorem 2.5 of [7] asserts that i... |

29 | Succinct representation of general unlabeled graphs
- Naor
- 1990
(Show Context)
Citation Context ... graphs; but note that the lower bound is d 64 4! e = 3. The standard adjacency list representation for G requires (n+2m) log n bits [1]. When m ? 3n 2 8 log n one can do better than this, as follows =-=[16]-=-: partition V into V 1 and V 2 of size n=2 each, and let G 1 and G 2 be the vertex-induced subgraphs on V 1 and V 2 , respectively. Represent G 1 and G 2 by adjacency matrices. Each vertex of V 1 or V... |

22 |
Representation of graphs
- Itai, Rodeh
- 1982
(Show Context)
Citation Context ...cupies O(n 2 ) space. A number of powerful connectivity algorithms exist for this representation. An adjacency list representation requires (n +m) log n space if the graph has m edges. Itai and Rodeh =-=[12]-=- showed that in certain cases this bound is tight. Deciding adjacency requiressO(log n) time, instead of O(1). (It should be noted that log n denotes the base two logarithm.) Finally, a graph can also... |

16 |
Four pages are necessary and sufficient for planar graphs
- Yannakakis
- 1986
(Show Context)
Citation Context ...in a book with O(g) pages in O(n + g) time. It is known that 4 pages are necessary and sufficient to embed planar graphs and that there exist graphs of arbitrarily high genus having 3-page embeddings =-=[19]-=-. It should be pointed out that the problem of finding an imbedding of G in a surface of minimal genus is intractable under the assumption that P 6= NP since Thomassen has shown that finding the genus... |

6 |
Pascal Graphs and Their Properties." The Fibonacci Quarterly 21
- Deo, Quinn
(Show Context)
Citation Context ...ve been constructed and studied, especially as the basis of parallel and distributed computing architectures. A survey of some of these structures is given in [11], and other examples may be found in =-=[6, 9]-=-. In this paper, we will look at two well-known graph properties namely genus and arboricity. Let NF (n) designate the number of graphs in F with n vertices. On average, any representation f must sati... |

5 |
Hinreichende Bedingungen fur die Existens von Teilgraphen, die zu einem vollstandigen Graphen homoomorph sind
- Mader
- 1972
(Show Context)
Citation Context ...s arboricity at most !H . Proof : Theorem 2.5 of [7] asserts that if G 2 F , then ms!H \Delta n. Actually, Theorem 2.5 deals with the notion of immersion, rather than minor inclusion, but a result in =-=[15]-=- shows that the assertion also applies to minor inclusion. The theorem now follows from the fact that every subgraph of G is also in F . 2 We now show how to apply the separator theorems just listed t... |

4 |
The page number of genus g graphs is o(g
- Heath, Istrail
- 1992
(Show Context)
Citation Context ...led pages. A book embedding of a graph is obtained by placing all the vertices on L and drawing each edge as a curve in one of the pages so that no two curves intersect. The following theorem is from =-=[10]-=-. Theorem 2 Given an imbedding in a surface, a graph G of genus g can be embedded in a book with O(g) pages in O(n + g) time. It is known that 4 pages are necessary and sufficient to embed planar grap... |

4 |
Interconnection networks and algorithms
- Hsu
- 1993
(Show Context)
Citation Context ...nnection patterns are known in advance have been constructed and studied, especially as the basis of parallel and distributed computing architectures. A survey of some of these structures is given in =-=[11]-=-, and other examples may be found in [6, 9]. In this paper, we will look at two well-known graph properties namely genus and arboricity. Let NF (n) designate the number of graphs in F with n vertices.... |

2 |
Compression of vertex transitive graphs. presented at 3rd Intl
- Litow
- 1997
(Show Context)
Citation Context ...morphism group. The family of vertex transitive graphs is a natural example of this kind of structure. Its subfamily of Cayley graphs, and special types of Cayley graphs have been widely studied. See =-=[2, 3, 14]-=- for further examples and references. Graphs whose interconnection patterns are known in advance have been constructed and studied, especially as the basis of parallel and distributed computing archit... |

1 | Fibonacci networks
- Govindaraju, Krishnamoorthy, et al.
- 1994
(Show Context)
Citation Context ...ve been constructed and studied, especially as the basis of parallel and distributed computing architectures. A survey of some of these structures is given in [11], and other examples may be found in =-=[6, 9]-=-. In this paper, we will look at two well-known graph properties namely genus and arboricity. Let NF (n) designate the number of graphs in F with n vertices. On average, any representation f must sati... |