## Obstructions for 2-Möbius band embedding extension problem (1997)

### Cached

### Download Links

- [www.fmf.uni-lj.si]
- [www.sfu.ca]
- [vega.fmf.uni-lj.si]
- DBLP

### Other Repositories/Bibliography

Venue: | SIAM J. Discrete Math |

Citations: | 3 - 3 self |

### BibTeX

@ARTICLE{Juvan97obstructionsfor,

author = {Martin Juvan and Bojan Mohar},

title = {Obstructions for 2-Möbius band embedding extension problem},

journal = {SIAM J. Discrete Math},

year = {1997},

volume = {10},

pages = {57--72}

}

### OpenURL

### Abstract

Abstract. Let K = C ∪ e1 ∪ e2 be a subgraph of G, consisting of a cycle C and disjoint paths e1 and e2, connecting two interlacing pairs of vertices in C. Suppose that K is embedded in the MöbiusbandinsuchawaythatC lies on its boundary. An algorithm is presented which in linear time extends the embedding of K to an embedding of G, if such an extension is possible, or finds a “nice ” obstruction for such embedding extensions. The structure of obtained obstructions is also analysed in details. Key words. surface embedding, obstruction, Möbius band, algorithm AMS subject classifications. 05C10, 05C85, 68Q20 1. Introduction. Let K be a subgraph of a graph G. A K-bridge (or a Kcomponent)inG is a subgraph of G which is either an edge e ∈ E(G)\E(K) (together with its endpoints) which has both endpoints in K, or it is a connected component of G − V (K) together with all edges (and their endpoints) between this component and K. EachedgeofaK-bridge B having an endpoint in K is a foot of B. The vertices

### Citations

307 |
Topological Graph Theory
- Gross, Tucker
- 1987
(Show Context)
Citation Context ...ended in [JM] and [M1]. Related results are also obtained in [M1, M2]. In our algorithms, we consider embeddings of graphs. In case of orientable surfaces, embeddings can be described combinatorially =-=[GT]-=- by specifying a rotation system: for each rotation system vertex v of the graph G we have cyclic permutationsv of its neighbors, representing their circular order around v on the surface. Although th... |

224 | Efficient planarity testing
- Hopcroft, Tarjan
- 1974
(Show Context)
Citation Context ...f a larger project [JMM, M4] which shows that there is a linear time algorithm to construct embeddings of graphs in an arbitrary (fixed) surface, generalizing the well-known Hopcroft-Tarjan algorithm =-=[HT]-=- for testing planarity in linear time. Our algorithms rely on the theory of bridges: a subgraph K of G is embedded in the surface and then this embedding is either extended to an embedding of G, or an... |

78 |
Time-bounded random access machines
- Cook, Reckhow
- 1972
(Show Context)
Citation Context ...atorial description. Concerning the time complexity of our algorithms, we assume a random-access machine (RAM) model with unit cost for basic operations. This model was introduced by Cook and Reckhow =-=[CR]-=-. More precisely, our model is the unit-cost RAM where operations on integers, whose value is O(n), need only constant time (n is the size of the given graph). 2. Parallel computations with constant t... |

8 |
Obstructions for the disk and the cylinder embedding extension problems
- Mohar
- 1994
(Show Context)
Citation Context ...n F ff [ F fi . It is clear by (M1) and (M3) that a thin millipede M obstructs embedding extensions of K to G. 4 Our notion of millipedes slightly differs from the concept of millipedes introduced in =-=[M2]-=-. The millipedes in [M2] can be shorter (i.e., m ! 7 is allowed) and their subgraphs B ffi i are allowed to be proper subgraphs of bridges in order that millipedes become minimal obstruction (with res... |

7 | Universal obstructions for embedding extension problems, Australas
- Mohar
(Show Context)
Citation Context ... graph is equivalent to the original one. 5 If B is a bridge of K in G, denote by b(B) the number of branches of B [ K that are contained in B. The number b(B) is called the size of B. size Lemma 4.1 =-=[M3]-=- Let G, K be as above. Every K-bridge B in G contains a subgraph ~ B with size at most 13 such that for an arbitrary set of non-local K-bridges B 1 ; :::; B k , any embedding of K [ ~ B 1 [ ::: [ ~ B ... |

4 | Efficient algorithm for embedding graphs in arbitrary surfaces - Juvan, Marincek, et al. |

3 | 2-restricted extensions of partial embeddings of graphs
- Juvan, Mohar
(Show Context)
Citation Context ...er small, or have a very special (millipede) structure. Moreover, finding an embedding extension or such an obstruction requires only linear time (Theorem 5.3). These results are used and extended in =-=[JM]-=- and [M1]. Related results are also obtained in [M1, M2]. In our algorithms, we consider embeddings of graphs. In case of orientable surfaces, embeddings can be described combinatorially [GT] by speci... |

2 | A linear time algorithm for embedding graphs in the torus, in preparation - Juvan, Marincek, et al. |

2 |
Projective plane and Mobius band obstructions, submitted
- Mohar
(Show Context)
Citation Context ... or have a very special (millipede) structure. Moreover, finding an embedding extension or such an obstruction requires only linear time (Theorem 5.3). These results are used and extended in [JM] and =-=[M1]-=-. Related results are also obtained in [M1, M2]. In our algorithms, we consider embeddings of graphs. In case of orientable surfaces, embeddings can be described combinatorially [GT] by specifying a r... |

2 |
A linear time algorithm for the 2-restricted embedding extension problem, in preparation
- Juvan, Mohar
(Show Context)
Citation Context ...er small, or have a very special (millipede) structure. Moreover, finding an embedding extension or such an obstruction requires only linear time (Theorem 5.3). These results are used and extended in =-=[JM]-=- and [M1]. Related results are also obtained in [M1, M2]. In our algorithms, we consider embeddings of graphs. In case of orientable surfaces, embeddings can be described combinatorially [GT] by speci... |

1 |
for the disk and the cylinder embedding extension problems
- Obstructions
- 1994
(Show Context)
Citation Context ...edded in Fα ∪ Fβ. It is clear by (M1) and (M3) that a thin millipede M obstructs embedding extensions of K to G. Our notion of millipedes slightly differs from the concept of millipedes introduced in =-=[M2]-=-. The millipedes in [M2] can be shorter (i.e., m<7 is allowed) and their subgraphs B◦ i are allowed to be proper subgraphs of bridges in order that millipedes become minimal obstruction (with respect ... |