## Graph Embedding with Topological Cycle-Constraints (1999)

Venue: | Graph Drawing, Proc. 7th Int. Symp. GD'99, LNCS 1731, 155--164 |

Citations: | 1 - 1 self |

### BibTeX

@INPROCEEDINGS{Dornheim99graphembedding,

author = {Christoph Dornheim},

title = {Graph Embedding with Topological Cycle-Constraints},

booktitle = {Graph Drawing, Proc. 7th Int. Symp. GD'99, LNCS 1731, 155--164},

year = {1999}

}

### OpenURL

### Abstract

. This paper concerns graph embedding under topological constraints. We address the problem of finding a planar embedding of a graph satisfying a set of constraints between its vertices and cycles that require embedding a given vertex inside its corresponding cycle. This problem turns out to be NP-complete. However, towards an analysis of its tractable subproblems, we develop an efficient algorithm for the special case where graphs are 2-connected and any two distinct cycles in the constraints have at most one vertex in common. 1 Introduction In many graph drawing applications it is important to find drawings with certain geometrical or topological properties. For instance, to support the semantics of visual languages, given subgraphs should be drawn with a predefined shape. Such user-defined requirements on the graph layout are called graph drawing constraints. Due to their wide applicability, graph drawing approaches supporting them are of recent research interest [7]. So far most o...

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Citation Context ...ected planar supergraph G 0 of G and verify in polynomial time G 0 to be S-embeddable. Next we show the problem to be NP-hard by transformation from the NPcomplete problem BETWEENNESS defined as (see =-=[4]-=-): given a finite set A and a set T of ordered triples of distinct elements from A, is there a one-to-one function f : A \Gamma! f1; : : : ; jAjg such that for each (a; b; c) 2 T either f(a) ! f(b) ! ... |

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Citation Context ...a representation of an S 0 -embedding. Roughly speaking, this algorithm is of the same spirit as the path addition procedure originally presented by Demoucron et al. [2] and also briefly described in =-=[1]-=-. As G and C r satisfy the necessary condition of Theorem 3, algorithm embedding generates a set F of cycles of G and a set F I of sets F I C ` F for each C 2 S such that for any F 2 F contained in no... |

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Citation Context ...ne vertex in common. 2 Preliminaries Assuming familiarity with basic graph terminology, we recall in this section definitions and properties of planar graphs and their embeddings as presented e.g. in =-=[3]-=-. Unless otherwise stated, all graphs are finite, simple and undirected. The set of vertices and edges of a graph G are denoted V (G) and E(G), respectively. A cycle C in G is induced if any edge of G... |

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Citation Context ...y Int C; ext C and Ext C are defined analogously. For characterizing graph embeddability with cycle-constraints we need the following notions and results concerning the cycle structure of graphs (see =-=[8]-=-, [3]). Given a graph G = (V; E) the edge space E(G) of G is the vector space over the 2-element field IF 2 = f0; 1g formed by the set of all subsets of E and the sum A \Phi B = (A n B) [ (B n A) for ... |

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Citation Context ...graph layout are called graph drawing constraints. Due to their wide applicability, graph drawing approaches supporting them are of recent research interest [7]. So far most of these approaches, e.g. =-=[5]-=-, focus on drawing arbitrary graphs with some sort of geometric constraints such as arithmetic restrictions on the coordinates of vertices. However, topological constraints that are less specific than... |

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Citation Context ...em 1 requires to determine the coefficients of a given cycle C with respect to a basis F of C(G 0 ) whose elements are inner facial cycles of some embedding f G 0 of G 0 . The following property (see =-=[6]-=-) leads directly to an appropriate procedure working also when G 0 is 2-connected. If FC is the subset of F whose elements sum up to C, the cycles of F n FC can be ordered as F 1 ; : : : ; F k such th... |

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Citation Context ...gorithm embedding given in Fig. 3 a representation of an S 0 -embedding. Roughly speaking, this algorithm is of the same spirit as the path addition procedure originally presented by Demoucron et al. =-=[2]-=- and also briefly described in [1]. As G and C r satisfy the necessary condition of Theorem 3, algorithm embedding generates a set F of cycles of G and a set F I of sets F I C ` F for each C 2 S such ... |

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Tamassia: Constraints in graph drawing algorithms
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Citation Context ...ape. Such user-defined requirements on the graph layout are called graph drawing constraints. Due to their wide applicability, graph drawing approaches supporting them are of recent research interest =-=[7]-=-. So far most of these approaches, e.g. [5], focus on drawing arbitrary graphs with some sort of geometric constraints such as arithmetic restrictions on the coordinates of vertices. However, topologi... |