## Contentment in graph theory: covering graphs with cliques (1977)

Citations: | 40 - 0 self |

### BibTeX

@MISC{Orlin77contentmentin,

author = {James Orlin},

title = {Contentment in graph theory: covering graphs with cliques},

year = {1977}

}

### Years of Citing Articles

### OpenURL

### Abstract

Fundamental questions posed by Boole in 1868 on the theory of sets have in recent years been translated to problems in graph theory. The major problems that this paper deals with are determining the minimum number of complete subgraphs of graph G which include all of the edges of G, and determining the minimum number of complete bipartite subgraphs which cover G. The two problems are of a very similar nature. Determining whether there is a projective plane of order p is a special case of the former problem. The latter problem has a natural translation into matrix theory which yields tight upper and lower bounds. An elementary proof is given for Graham's theorem. Two non-obvious classes are given for which the above problems are easily handled; however, this author doubts that these classes can be extended significantly. Two new problems are shown in this

### Citations

502 |
Reducibility among combinatorial problems. Complexity of computer computations
- Karp
- 1972
(Show Context)
Citation Context ...d a translation of Remark 5.2 to directed graphs. Section 8: NP-COMPLETENESS There has been much research in recent years on a class of problems called VNP-complete. The reader is referred to [3] and =-=[9]-=- for the exact definitions and further information on NP-completeness. Briefly and inprecisely a problemn P is said to be reducible to another problem Q if it is possible to translate problem P in a p... |

35 | The representation of graphs by set intersections
- Erdò‹s, Goodman, et al.
- 1966
(Show Context)
Citation Context ...ces x, ... , x,. Then there exists a set S and a family of subsets Si, .. , Sn of S such that Iz n SjI is equal to the number of edges in joining to xj. A problem posed by Erd6s, Goodman, and Posa in =-=[4]-=- and motivated by Boole in [1] is the following: what is the minimum number of elements in a set S that satisfies Theorem 1.1 ? Boole referred to the number as the content of the system of sets. It is... |

16 |
Some properties of line digraphs
- Harary, Norman
- 1960
(Show Context)
Citation Context ... edge directed in and at least one edge directed out. Let T(D) be the number of distinct through vertices in D. THEORE '7.2: B(D*) = B(D*) =T(D). PROOF. According to a theorem of Harary and Norman in =-=[8]-=- D has the property thatj every edge belongs to a unique maximal DCBS, and that this DCBS is induced by a through vertex. Thus Theorem 7.2 follows from this and a translation of Remark 5.2 to directed... |

6 |
The Steiner Problem
- Foulds, Graham
- 1982
(Show Context)
Citation Context ...If A is symmnunetric, then all eigenvalues are real and A has full geometric multiplicity. Hence, in this case N+(A)+N-(A)= rank (A). The following theorem is due to H. S. Witsenhausen but appears in =-=[6]-=-. THEOREM 7.2: B(G) > max (N+(M0 ), N-(MG)). LErMMs 7.3: Let A be a real symmetric matrix. Let V + and V- denote respectively the vector space generated by the eigenvectors corresponding to positive e... |

6 |
On coverings of graphs, in Theory of Graphs
- LovAsz
- 1968
(Show Context)
Citation Context ...mum nu.mber of elements in the set S is equal to the R-content of G. Upper bounds for the content and R-corintent were given by Erdds, Goodman and Posa in [4], by M. Hall Jr. in [5], and by Lovasz in =-=[12]-=-. The problem was also considered by Ryser in [13]. Section 2: SOME ELEMENTARY REMARKS ON THE CONTENT AND R-CONTENT REMARK 2.1: If G is a graph and W is a subset of the set of vertices of G, then C(G)... |

5 |
S.: Boolean Methods in Operations Research
- Hammer, Rudeanu
- 1968
(Show Context)
Citation Context ...e corollary then follows from induction based on Lemma 3.5. UNSOLVED PROBLEM: What is the asymptotic behavior of RC(T 2 n)? I make the following remark without proof. For explanation of the terms see =-=[7]-=-. REMARK 3.7: Let S = class of Boolean functions f: {0, l}n -t {0, 1} with the property that all of the prime implicants have length n. Let S' be the subclass of S consisting of functions with the add... |

2 |
On a combinatorial problem
- BRUIJN, P
- 1948
(Show Context)
Citation Context ...l; C1=G [Type 1] 2) r n; one of the cliques contains n-1I vertices. The remaining n-1 cliques are edges. [Type 2] 3) r = n and all the cliques are of size k + where 2 + k k- 1= n. [Type 3] PROOF. See =-=[2]-=- for alternate formulation and proof. REMARK 3.1: It is not difficult to show that a decomposition of type 3 corresponds to a projective plane. The cliques represent lines and the vertices represent p... |

1 |
Of Propositions Numerically Definite, 1868
- Boole
- 1952
(Show Context)
Citation Context ...ists a set S and a family of subsets Si, .. , Sn of S such that Iz n SjI is equal to the number of edges in joining to xj. A problem posed by Erd6s, Goodman, and Posa in [4] and motivated by Boole in =-=[1]-=- is the following: what is the minimum number of elements in a set S that satisfies Theorem 1.1 ? Boole referred to the number as the content of the system of sets. It is no coincidence that the conte... |

1 |
The Complexity of Theorem Proving
- Cooke
- 1970
(Show Context)
Citation Context ... this and a translation of Remark 5.2 to directed graphs. Section 8: NP-COMPLETENESS There has been much research in recent years on a class of problems called VNP-complete. The reader is referred to =-=[3]-=- and [9] for the exact definitions and further information on NP-completeness. Briefly and inprecisely a problemn P is said to be reducible to another problem Q if it is possible to translate problem ... |

1 |
A Problem in Partitions
- Jr, M
- 1941
(Show Context)
Citation Context ...are adjacent. The minimum nu.mber of elements in the set S is equal to the R-content of G. Upper bounds for the content and R-corintent were given by Erdds, Goodman and Posa in [4], by M. Hall Jr. in =-=[5]-=-, and by Lovasz in [12]. The problem was also considered by Ryser in [13]. Section 2: SOME ELEMENTARY REMARKS ON THE CONTENT AND R-CONTENT REMARK 2.1: If G is a graph and W is a subset of the set of v... |

1 | Theorie der Endlichen und Unendlichen Graphen - K6nig - 1950 |

1 |
Wong - The Complexity of the Minimal Edge Clique Cover Problem with Application to Keyword
- Kou, Stockmeyer, et al.
- 1976
(Show Context)
Citation Context ...bvious that determining the content or bicontent of a graph is NP-complete. However, the author knows of no proof of the assertion. It is known that both problems are in the class NP. It was prove in =-=[11]-=- that problem P3 (and hence also P4) is NP-complete. That publication appeared after the original writing of this paper and was discovered independently. The proofs are similar in nature although far ... |

1 |
Intersection Properties of Finite Sets
- Ryser
- 1973
(Show Context)
Citation Context ...he R-content of G. Upper bounds for the content and R-corintent were given by Erdds, Goodman and Posa in [4], by M. Hall Jr. in [5], and by Lovasz in [12]. The problem was also considered by Ryser in =-=[13]-=-. Section 2: SOME ELEMENTARY REMARKS ON THE CONTENT AND R-CONTENT REMARK 2.1: If G is a graph and W is a subset of the set of vertices of G, then C(G) > C(GI W) and also RC(C7) > R(1 W).408 PROOF. Le... |

1 | Sur deux properietes des classes d'ensembles, Fund - Szpilrajn-Marczkrwski - 1945 |