## Adjacency posets of planar graphs

Venue: | DISCRETE MATH |

Citations: | 4 - 3 self |

### BibTeX

@ARTICLE{Felsner_adjacencyposets,

author = {Stefan Felsner and Ching Man Li and William T. Trotter},

title = {Adjacency posets of planar graphs},

journal = {DISCRETE MATH},

year = {},

pages = {1097--1104}

}

### OpenURL

### Abstract

In this paper, we show that the dimension of the adjacency poset of a planar graph is at most 8. From below, we show that there is a planar graph whose adjacency poset has dimension 5. We then show that the dimension of the adjacency poset of an outerplanar graph is at most 5. From below, we show that there is an outerplanar graph whose adjacency poset has dimension 4. We also show that the dimension of the adjacency poset of a planar bipartite graph is at most 4. This result is best possible. More generally, the dimension of the adjacency poset of a graph is bounded as a function of its genus and so is the dimension of the vertex-face poset of such a graph.

### Citations

201 |
Embedding planar graphs on the grid
- Schnyder
- 1990
(Show Context)
Citation Context ...r’s proof of Theorem 1.1 is a special coloring and orientation of the interior edges of a triangulation, today known as Schnyder wood. For existence and the theory of Schnyder woods we refer to [12], =-=[13]-=-, [9] and [11]. Below we collect some of the features of Schnyder woods needed in our context. Schnyder Paths and Regions Let T be a planar triangulation in which the three exterior vertices are label... |

165 |
Combinatorics and partially ordered sets: Dimension theory, Johns Hopkins Series in the Mathematical Sciences
- Trotter
- 1992
(Show Context)
Citation Context ...t dg. Our presentation will require a few well known tools from dimension theory, and we briefly summarize these results here. For additional background material, we refer the reader to the monograph =-=[14]-=- and the survey paper [15]. 1.2. Background Material on Posets. When P is a poset, we let Inc(P) denote the set of all incomparable pairs of P. When (x, y) ∈ Inc(P) and L is a linear extension of P, w... |

78 |
Partially ordered sets
- Trotter
- 1995
(Show Context)
Citation Context ...l require a few well known tools from dimension theory, and we briefly summarize these results here. For additional background material, we refer the reader to the monograph [14] and the survey paper =-=[15]-=-. 1.2. Background Material on Posets. When P is a poset, we let Inc(P) denote the set of all incomparable pairs of P. When (x, y) ∈ Inc(P) and L is a linear extension of P, we say (x, y) is reversed i... |

34 | Convex drawings of planar graphs and the order dimension of 3-polytopes
- Felsner
(Show Context)
Citation Context ...awn on a surface without crossings, then we may also consider the vertex-edge-face poset, a poset of height 3. The following theorem is due to Brightwell and Trotter [4]. Simpler proofs were given in =-=[7]-=-, [8] and [11]. Theorem 1.2. If a planar 3-connected graph G is drawn without edge crossings in the plane, then the dimension of the vertex-edge-face poset is 4. Furthermore, if any vertex or any face... |

26 | On acyclic colorings of graphs on surfaces
- Alon, Mohar, et al.
- 1994
(Show Context)
Citation Context ...coloring of G using t colors so that for every two colors, the subgraph of G induced by the vertices assigned these colors is acyclic. The next theorem is due to Albertson and Bermen [1]. Alon et al. =-=[2]-=- have estimated the bound ag as O((2g − 2) 4/7 ). Theorem 8.2. For every non-negative integer g, there exists an integer ag so that if G is a graph of genus g, then the acyclic chromatic number of G i... |

20 | Geometric graphs and arrangements
- Felsner
- 2004
(Show Context)
Citation Context ...oof of Theorem 1.1 is a special coloring and orientation of the interior edges of a triangulation, today known as Schnyder wood. For existence and the theory of Schnyder woods we refer to [12], [13], =-=[9]-=- and [11]. Below we collect some of the features of Schnyder woods needed in our context. Schnyder Paths and Regions Let T be a planar triangulation in which the three exterior vertices are labeled v0... |

19 |
The order dimension of convex polytopes
- Brightwell, Trotter
- 1993
(Show Context)
Citation Context ... at most three. When a graph is drawn on a surface without crossings, then we may also consider the vertex-edge-face poset, a poset of height 3. The following theorem is due to Brightwell and Trotter =-=[4]-=-. Simpler proofs were given in [7], [8] and [11]. Theorem 1.2. If a planar 3-connected graph G is drawn without edge crossings in the plane, then the dimension of the vertex-edge-face poset is 4. Furt... |

19 |
Planar graphs and poset dimension, Order 5
- Schnyder
- 1989
(Show Context)
Citation Context ...is less than an edge e in the incidence poset of the graph when x is one of the two endpoints of e. Interest in incidence posets was initiated with the following remarkable theorem due to W. Schnyder =-=[12]-=-. Theorem 1.1. A graph is planar if and only if the dimension of its incidence poset is at most three. When a graph is drawn on a surface without crossings, then we may also consider the vertex-edge-f... |

18 |
An acyclic analogue to Heawood’s theorem
- Albertson, Berman
- 1978
(Show Context)
Citation Context ...here is a proper coloring of G using t colors so that for every two colors, the subgraph of G induced by the vertices assigned these colors is acyclic. The next theorem is due to Albertson and Bermen =-=[1]-=-. Alon et al. [2] have estimated the bound ag as O((2g − 2) 4/7 ). Theorem 8.2. For every non-negative integer g, there exists an integer ag so that if G is a graph of genus g, then the acyclic chroma... |

11 | The order dimension of planar maps
- Brightwell, Trotter
- 1997
(Show Context)
Citation Context ...nus. Stefan Felsner is partially supported by DFG grant FE-340/7-1. Ching Li is supported by the Croucher Foundation and Berlin Mathematical School. 12 FELSNER, LI, AND TROTTER In a subsequent paper =-=[5]-=-, Brightwell and Trotter extended the preceding theorem with the following result for planar graphs with loops and multiple edges allowed. Theorem 1.3. If a planar multigraph is drawn without crossing... |

11 |
The dimension of planar posets
- Trotter, Moore
- 1977
(Show Context)
Citation Context ...-negative integer g, there exists an integer ag so that if G is a graph of genus g, then the acyclic chromatic number of G is at most ag. Second, we have the following result due to Trotter and Moore =-=[16]-=-. Theorem 8.3. Let P be a poset whose diagram is a tree (or a forest). Then the dimension of P is at most 3. Note that the diagram of the poset Q shown in Figure 4 is a tree and it has dimension 3, so... |

8 |
On the complexity of posets
- Trotter, Bogart
- 1976
(Show Context)
Citation Context ...aximal element is not empty. The minimum number of linear extensions reversing S is the interval dimension of P. This parameter was defined for general posets (arbitrary height) by Bogart and Trotter =-=[3]-=-. There are posets of large dimension and small interval dimansion, this remains true for height 2. We go for a contradiction. Suppose that there is a family F = {L1, L2, L3} reversing all critical pa... |

6 | Schnyder woods and orthogonal surfaces
- Zickfeld, Felsner
- 2006
(Show Context)
Citation Context ...ace without crossings, then we may also consider the vertex-edge-face poset, a poset of height 3. The following theorem is due to Brightwell and Trotter [4]. Simpler proofs were given in [7], [8] and =-=[11]-=-. Theorem 1.2. If a planar 3-connected graph G is drawn without edge crossings in the plane, then the dimension of the vertex-edge-face poset is 4. Furthermore, if any vertex or any face is removed, t... |

2 |
Geodesic Embeddings and Planar Graphs Order 20
- Felsner
- 2003
(Show Context)
Citation Context ...n a surface without crossings, then we may also consider the vertex-edge-face poset, a poset of height 3. The following theorem is due to Brightwell and Trotter [4]. Simpler proofs were given in [7], =-=[8]-=- and [11]. Theorem 1.2. If a planar 3-connected graph G is drawn without edge crossings in the plane, then the dimension of the vertex-edge-face poset is 4. Furthermore, if any vertex or any face is r... |

1 |
3-interval irreducible partially ordered sets, Order 11
- Felsner
- 1994
(Show Context)
Citation Context ...t include the easy proof but remark that the claim is equivalent to the statement that Q has interval dimension 3. Moreover the removal of any point from Q lowers the interval dimension to 2. Felsner =-=[6]-=-, has characterized all posets of height 2 with this property. With the claim we can complete the proof. Note that the elements in {x′ : φ(x) = 3} ∪ {y′′ : φ(y) = 2} form a copy of the poset Q shown i... |

1 |
Dimension, graph and hypergraph coloring, Order 17
- Felsner, Trotter
- 2000
(Show Context)
Citation Context ...ies that there are graphs of arbitrary genus whose incidence posets have bounded dimension. 1.1. Adjacency Posets. Motivated by connections between chromatic number and poset dimension we proposed in =-=[10]-=- to investigate the adjacency poset of a finite simple 1 graph G. This poset P has V ′ ∪ V ′′ as its set of points where V ′ = {v ′ : v ∈ V } and V ′′ = {v ′′ : v ∈ V } are copies of the vertex set V ... |