## Planarization of Graphs Embedded on Surfaces (1995)

Venue: | in WG |

Citations: | 6 - 1 self |

### BibTeX

@INPROCEEDINGS{Djidjev95planarizationof,

author = {Hristo N. Djidjev and Shankar M. Venkatesan},

title = {Planarization of Graphs Embedded on Surfaces},

booktitle = {in WG},

year = {1995},

pages = {62--72},

publisher = {Springer}

}

### OpenURL

### Abstract

A planarizing set of a graph is a set of edges or vertices whose removal leaves a planar graph. It is shown that, if G is an n-vertex graph of maximum degree d and orientable genus g, then there exists a planarizing set of O( p dgn) edges. This result is tight within a constant factor. Similar results are obtained for planarizing vertex sets and for graphs embedded on nonorientable surfaces. Planarizing edge and vertex sets can be found in O(n + g) time, if an embedding of G on a surface of genus g is given. We also construct an approximation algorithm that finds an O( p gn log g) planarizing vertex set of G in O(n log g) time if no genus-g embedding is given as an input. 1 Introduction A graph G is planar if G can be drawn in the plane so that no two edges intersect. Planar graphs arise naturally in many applications of graph theory, e.g. in VLSI and circuit design, in network design and analysis, in computer graphics, and is one of the most intensively studied class of graphs [2...

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Citation Context ... Foundation grant CCR-9409191. edges, there is no guarantee that that edge set is of small size. On the other hand the problem of finding a minimum planarizing set of edges is known to be NP-complete =-=[11]-=-. In this paper we show that for any n vertex graph of bounded genus g and maximum degree d there exists a planarizing edge set of size O( p dgn). This result is tight within a constant factor and imp... |

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Citation Context ...oof of Theorem 3, there exists a planarizing edge set of size not exceeding 4 p dgm 0 for G. Then 4 p dgm 0 = 4 q dg(m \Gamma jU G t j)s4 q 2dg(n + 2g \Gamma 2): The skewness of a graph is defined in =-=[15]-=- as the smallest number of edges whose removal leads to a planar graph. Then by Theorem 4 the skewness of the class of n-vertex g-genus d-degree graphs is not exceeding 4 p 2dg(n + 2g \Gamma 2). 4 Ext... |

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Citation Context ...f degree d and genus g [7, 13, 26]. If one can planarize a graph by removing o( p gn) vertices or o( p dgm) edges, then one can then find in the resulting planar graph a separator of O( p n) vertices =-=[20]-=- or O( p dn) edges [4], which will be a contradiction to the above lower bounds. For graphs embedded on nonorientable surfaces the tightness of our results follows from the tightness in the orientable... |

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Citation Context ... describe an efficient algorithm that constructs a planarizing set without knowing the embedding of the graph on its genus surface. 2 Preliminaries By a surface, we mean a closed connected 2-manifold =-=[14, 12]-=-. There are two major types of surfaces: orientable and nonorientable surfaces. Informally, if an intelligent bug starts from some point on a closed curve drawn on a surface, traverses the curve, and ... |

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Citation Context ...arise naturally in many applications of graph theory, e.g. in VLSI and circuit design, in network design and analysis, in computer graphics, and is one of the most intensively studied class of graphs =-=[23]-=-. Many problems that are computationally hard for arbitrary graphs have efficient solutions for the case of planar graphs. Examples include minimum spanning trees, network flow problems, shortest path... |

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Citation Context ... in the case where an embedding of the graph on its genus surface is not given. This is motivated by the fact that the problem of embedding a graph on a surface of minimum genus is known to be NPhard =-=[27]-=-. The best known algorithm for the latter problem is polynomial on the number of vertices n, but doubly exponential on the genus g [10]. In this paper we describe an approximation algorithm that finds... |

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Citation Context ...close to the given graph, as possible. A problem of this type is called a graph planarization problem. This problem has been intensively investigated in relation to its applications to circuit layout =-=[21, 3, 24, 18]-=-. One approach to the graph planarization problem is to construct a maximal planar subgraph of the input graph G. By solving the maximal planar subgraph problem one finds a minimal set of edges whose ... |

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Citation Context ...he root of the spanning tree of the corresponding component of G 00 and thus jP K 00 js4g K 00 (r \Gamma 2) + 1. Since the genus of any graph is equal to sum of the genera of its connected components =-=[1]-=-, then g 00 is equal to the sum of g K 00 over all components K 00 of G 00 . From this fact and (3) applied to all components K of G 0 we have g 00 \Gamma g 0sd \Gamma k 0 + 1; and by Lemma 3.1 g 0 + ... |

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Citation Context ...eaves a planar graph (called a planarizing set of edges). Several fast algorithms have been recently proposed for the maximal planar subgraph problem, including the O(m log n) algorithm of Cai et al. =-=[2]-=-, the O(m+nff(m; n)) algorithm of La Poutr'e [25] and the O(m+n) algorithm of Djidjev [9] (n and m are the number of vertices and edges respectively). Although a solution of the maximal planar subgrap... |

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Citation Context ...close to the given graph, as possible. A problem of this type is called a graph planarization problem. This problem has been intensively investigated in relation to its applications to circuit layout =-=[21, 3, 24, 18]-=-. One approach to the graph planarization problem is to construct a maximal planar subgraph of the input graph G. By solving the maximal planar subgraph problem one finds a minimal set of edges whose ... |

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Citation Context ...ding a graph on a surface of minimum genus is known to be NPhard [27]. The best known algorithm for the latter problem is polynomial on the number of vertices n, but doubly exponential on the genus g =-=[10]-=-. In this paper we describe an approximation algorithm that finds an O( p dgn log g) planarizing edge set that does not require a genus-g embedding to be given as an input. No comparable algorithm for... |

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Citation Context ...close to the given graph, as possible. A problem of this type is called a graph planarization problem. This problem has been intensively investigated in relation to its applications to circuit layout =-=[21, 3, 24, 18]-=-. One approach to the graph planarization problem is to construct a maximal planar subgraph of the input graph G. By solving the maximal planar subgraph problem one finds a minimal set of edges whose ... |

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Citation Context ...f edges). Several fast algorithms have been recently proposed for the maximal planar subgraph problem, including the O(m log n) algorithm of Cai et al. [2], the O(m+nff(m; n)) algorithm of La Poutr'e =-=[25]-=- and the O(m+n) algorithm of Djidjev [9] (n and m are the number of vertices and edges respectively). Although a solution of the maximal planar subgraph problem defines a minimal planarizing set of Th... |

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Citation Context ...section we show that almost optimal planarizing vertex set (upto a factor of O( p log g)) can be found for any n-vertex g-genus graph in O(n log g) time. We will make use of the following result from =-=[8]-=-. Theorem 7 For any n vertex graph G a partitioning A, B, C of the vertices of G can be found in O(n) time such that no edge joins a vertex in A with a vertex in B, jAj,jBjsn=2, jCjsc p (g 0 + 1)n, an... |

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Citation Context ...en recently proposed for the maximal planar subgraph problem, including the O(m log n) algorithm of Cai et al. [2], the O(m+nff(m; n)) algorithm of La Poutr'e [25] and the O(m+n) algorithm of Djidjev =-=[9]-=- (n and m are the number of vertices and edges respectively). Although a solution of the maximal planar subgraph problem defines a minimal planarizing set of This work is partially supported by Nation... |

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Citation Context ...h of any fundamental cycle is at most 2r + 1, if it contains t, or at most 2r \Gamma 1, otherwise. We are going to use the following known topological facts (for simple combinatorial proofs see, e.g. =-=[7, 13]-=-.) Lemma 2.1 Let c be a noncontractible nonseparating curve on a surface S. Then S \Gamma c can be embedded on a surface of characteristic X(S)+ 2, if c is orientation preserving, or X(S)+ 1, if c is ... |

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Citation Context ...[7, 13, 26]. If one can planarize a graph by removing o( p gn) vertices or o( p dgm) edges, then one can then find in the resulting planar graph a separator of O( p n) vertices [20] or O( p dn) edges =-=[4]-=-, which will be a contradiction to the above lower bounds. For graphs embedded on nonorientable surfaces the tightness of our results follows from the tightness in the orientable case. Let G be any n-... |

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Citation Context ...ious proofs are quite complex and the constants are large. We present here very simple proof that also gives a leading constant 4, improving the previous leading constants 44= p 3 of [5, 6] and 26 of =-=[17]-=-. We consider also graphs embedded on nonorientable surfaces, showing that similar bounds hold for the sizes of the smallest edge and vertex planarizing sets. Our proof technique combines a careful ex... |

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Citation Context ...lar bounds hold for the sizes of the smallest edge and vertex planarizing sets. Our proof technique combines a careful examination of the topology of the graph with a use of a radius reduction device =-=[28]-=-. Our proofs are constructive, giving linear algorithms that find the planarizing sets, if an embedding of the graph on its genus surface is given. We also investigate the problem of finding a planari... |

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Citation Context |

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On Some Properties of Nonplanar Graphs
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Citation Context ...ex graph of bounded genus g and maximum degree d there exists a planarizing edge set of size O( p dgn). This result is tight within a constant factor and improves the best previous bound of O(d p gn) =-=[5]-=-. We also consider the related problem of finding a small planarizing set of vertices. Although an asymptotically optimal O( p gn) bound for this problem is known [5], the previous proofs are quite co... |

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Citation Context ... describe an efficient algorithm that constructs a planarizing set without knowing the embedding of the graph on its genus surface. 2 Preliminaries By a surface, we mean a closed connected 2-manifold =-=[14, 12]-=-. There are two major types of surfaces: orientable and nonorientable surfaces. Informally, if an intelligent bug starts from some point on a closed curve drawn on a surface, traverses the curve, and ... |

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Citation Context ...h of any fundamental cycle is at most 2r + 1, if it contains t, or at most 2r \Gamma 1, otherwise. We are going to use the following known topological facts (for simple combinatorial proofs see, e.g. =-=[7, 13]-=-.) Lemma 2.1 Let c be a noncontractible nonseparating curve on a surface S. Then S \Gamma c can be embedded on a surface of characteristic X(S)+ 2, if c is orientation preserving, or X(S)+ 1, if c is ... |

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Citation Context ...own [5], the previous proofs are quite complex and the constants are large. We present here very simple proof that also gives a leading constant 4, improving the previous leading constants 44= p 3 of =-=[5, 6]-=- and 26 of [17]. We consider also graphs embedded on nonorientable surfaces, showing that similar bounds hold for the sizes of the smallest edge and vertex planarizing sets. Our proof technique combin... |

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Citation Context ...hs with no more than n i =2 vertices each, by removing a set C i of O( p (g i + 1)n i ) vertices of G i , where g i and n i are the genus and the number of vertices of G i . 5. Use the algorithm from =-=[16]-=- to test each subgraph for planarity. 6. Let C := C [ C i . Add the nonplanar subgraphs found in Step 5 to M . Let us estimate the maximum time T (n; g) and the maximum size S(n; g) of the set C found... |

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Citation Context ... by the set of vertices at level l i . Consider the subgraph L 0 of L induced by the set of the vertices of L adjacent to vertices at level l i + 1. Then L 0 is a union of edge disjoint simple cycles =-=[22]-=- implying that L contains at least one simple cycle. Denote by c one such simple cycle. Delete all edges of EM incident to vertices of c. Let G 0 be the resulting graph and k c and g c be the number o... |

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Citation Context ...ess in the orientable case follows from the fact that\Omega\Gamma p gn) vertices or \Omega\Gamma p dgm) edges are needed in the worst case to separate an n-vertex m-edge graph of degree d and genus g =-=[7, 13, 26]-=-. If one can planarize a graph by removing o( p gn) vertices or o( p dgm) edges, then one can then find in the resulting planar graph a separator of O( p n) vertices [20] or O( p dn) edges [4], which ... |