## A Note on Linear Discrepancy and Bandwidth (2002)

Venue: | J. Combin. Math. Combin. Comput |

Citations: | 7 - 0 self |

### BibTeX

@ARTICLE{Rautenbach02anote,

author = {Dieter Rautenbach},

title = {A Note on Linear Discrepancy and Bandwidth},

journal = {J. Combin. Math. Combin. Comput},

year = {2002},

volume = {55},

pages = {199--208}

}

### OpenURL

### Abstract

Fishburn, Tanenbaum and Trenk [4] de ne the linear discrepancy ld(P ) of a poset P = (V; < P ) as the minimum integer k 0 for which there exists a bijection f : V ! f1; 2; : : : ; jV jg such that u < P v implies f(u) < f(v) and ujj P v implies jf(u) f(v)j k. In [5] they prove that the linear discrepancy of a poset equals the bandwidth of its cocomparability graph.

### Citations

116 | A survey on graph layout problems
- Díaz, Petit, et al.
(Show Context)
Citation Context ... v implies jf(u) f(v)j k. Let G = (V; E) be a graph. A bijective mapping f : V ! f1; 2; : : : ; jV jg such that uv 2 E implies jf(u) f(v)j k, is called a k-labeling of G. The bandwidth bw(G) (cf. [3=-=]-=-) of G is the minimum k for which there exists a k-labeling of G. The main result of [5] relates the linear discrepancy to the bandwidth. Theorem 1 (Fishburn, Tanenbaum and Trenk [5]) If P is a poset ... |

73 | Complexity results for bandwidth minimization
- Garey, Graham, et al.
- 1978
(Show Context)
Citation Context ...l now turn our attention to Problem (iii). It is well-known that graphs with bandwidth at most 2 can be recognized in linear time and that a 2-labeling of such graphs can also be found in linear time =-=[6]-=-,[9],[1]. In view of Theorem 1, this implies that posets with linear discrepancy at most 2 can be recognized in linear time. Furthermore, we will describe now how the proof of Theorem 1 in [5] implies... |

67 |
Tree-width and tangles: an new connectivity measure and some applications
- Reed
- 1997
(Show Context)
Citation Context ... Problem (ii) is (G) = 3. In fact, there are planar graphs of maximum degree 3 that have arbitrarily large bandwidth (consider e.g. the so-called walls that even have arbitrarily large treewidth cf. [=-=-=-10]). Therefore, (G) = 3 is also thesrst case where the assumption that the graph is a cocomparability graph has to play some role. In view of Theorem 1, Proposition 1 immediately implies bw(G) 2(G) ... |

35 | Approximating the bandwidth for asteroidal triple-free graphs
- Kloks, Kratsch, et al.
- 1999
(Show Context)
Citation Context ...us j i 2(G) 1 which implies (1). The inequalities (2) and (3) follow immediately from two known lower bounds on the bandwidth: bw(G) max n 1 3 (jN G (u) [ NG (v)j 1) j uv 2 E o (cf. Lemma 2.3 in [8]) and bw(G) d (G) 2 e (cf. [2] or Lemma 18 in [4]). Q.E.D. We will illustrate that Proposition 1 is best-possible. For l 0 let P = (V;sP ) be the poset such that V = fx; yg [ fu 1 ; u 2 ; : : : ;... |

32 |
A remark on a problem of
- Chvátal
- 1970
(Show Context)
Citation Context ...). The inequalities (2) and (3) follow immediately from two known lower bounds on the bandwidth: bw(G) max n 1 3 (jN G (u) [ NG (v)j 1) j uv 2 E o (cf. Lemma 2.3 in [8]) and bw(G) d (G) 2 e (cf. [2] or Lemma 18 in [4]). Q.E.D. We will illustrate that Proposition 1 is best-possible. For l 0 let P = (V;sP ) be the poset such that V = fx; yg [ fu 1 ; u 2 ; : : : ; u 2l+2 g, u isP u j for 1 is ... |

27 |
Computing the Bandwidth of Interval Graphs
- Kleitman, Vohra
- 1990
(Show Context)
Citation Context ...andwidth-2 problem for an interval graph G 0 . Given P and the 2-labeling of G, the graph G 0 can be eciently constructed and a special 2-labeling of G 0 is found using the linear time algorithm from =-=[7]-=-. Using the original 2labeling of G and the special 2-labeling of G 0 , it is then possible to obtain in polynomial time the desired linear extension of P using a so-called Switching Lemma (cf. Lemma ... |

17 |
Linear discrepancy and weak discrepancy of partially ordered sets
- Tanenbaum, Trenk, et al.
(Show Context)
Citation Context ...merly Bellcore). DIMACS was founded as an NSF Science and Technology Center, and also receives support from the New Jersey Commission on Science and Technology. ABSTRACT Fishburn, Tanenbaum and Trenk =-=[-=-4] dene the linear discrepancy ld(P ) of a poset P = (V;sP ) as the minimum integer k 0 for which there exists a bijection f : V ! f1; 2; : : : ; jV jg such that usP v implies f(u)sf(v) and ujj P v i... |

6 | On bandwidth-2 graphs
- Caprara, Malucelli, et al.
(Show Context)
Citation Context ...rn our attention to Problem (iii). It is well-known that graphs with bandwidth at most 2 can be recognized in linear time and that a 2-labeling of such graphs can also be found in linear time [6],[9],=-=[1]-=-. In view of Theorem 1, this implies that posets with linear discrepancy at most 2 can be recognized in linear time. Furthermore, we will describe now how the proof of Theorem 1 in [5] implies that a ... |

4 |
A simple linear-time algorithm for the recognition of bandwidth-2 biconnected garphs
- Makedon, Sheinwald, et al.
- 1993
(Show Context)
Citation Context ...w turn our attention to Problem (iii). It is well-known that graphs with bandwidth at most 2 can be recognized in linear time and that a 2-labeling of such graphs can also be found in linear time [6],=-=[9]-=-,[1]. In view of Theorem 1, this implies that posets with linear discrepancy at most 2 can be recognized in linear time. Furthermore, we will describe now how the proof of Theorem 1 in [5] implies tha... |

2 |
Linear discrepancy and bandwidth, Order 18
- Fishburn, Tanenbaum, et al.
- 2001
(Show Context)
Citation Context ...ar time [6],[9],[1]. In view of Theorem 1, this implies that posets with linear discrepancy at most 2 can be recognized in linear time. Furthermore, we will describe now how the proof of Theorem 1 in =-=[5]-=- implies that a linear extension of uncertainty at most 2 of such posets can be found in polynomial time. Given a 2-labeling of the cocomparability graph G of a poset P (which can be found in linear t... |