Embedding Graphs Simultaneously with Fixed Edges

Frati, Fabrizio

doi:10.1007/978-3-540-70904-6_12

Cited by 28 publications

(33 citation statements)

References 9 publications

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“…Erten and Kobourov [3] showed that any pair of a tree and a path always has a simultaneous embedding with fixed edges. Di Giacomo and Liotta [2] extended this result by showing that any pair of an outerplanar graph with a cycle has a simultaneous embedding with fixed edges while Frati [6] showed that any pair of a planar graph and a tree has a simultaneous embedding with fixed edges. Fowler et al [5] used Frati's result as a starting point to characterize the set of planar graphs that have a simultaneous embedding with fixed edges with any planar graph in two ways: by a forbidden minor and by a complete list of graphs with this property.…”

Section: Introductionmentioning

confidence: 99%

Intersection Graphs in Simultaneous Embedding with Fixed Edges

Jünger¹,

Schulz²

2009

JGAA

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Abstract. We examine the simultaneous embedding with fixed edges problem for two planar graphs G1 and G2 with the focus on their intersection S := G1 ∩ G2. In particular, we will present the complete set of intersection graphs S that guarantee a simultaneous embedding with fixed edges for (G1, G2). More formally, we define the subset I SEFE of all planar graphs as follows: A graph S lies in I SEFE if every pair of planar graphs (G1, G2) with intersection S = G1 ∩ G2 has a simultaneous embedding with fixed edges. We will characterize this set by a detailed study of topological embeddings and finally give a complete list of graphs in this set as our main result of this paper.

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Section: Introductionmentioning

confidence: 99%

Intersection Graphs in Simultaneous Embedding with Fixed Edges

Jünger¹,

Schulz²

2009

JGAA

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“…Many approaches have been made to decide the problem for some classes of graph pairs [4,7,9,10]. Frati [10] showed that trees and planar graphs always have a SEFE.…”

Section: Introductionmentioning

confidence: 99%

“…Frati [10] showed that trees and planar graphs always have a SEFE. Fowler et al [9] improved this result to show that forests, circular caterpillars (removal of all degree-1 vertices yields a cycle), K 4 , and subgraphs of K 3 -multiedges (an edge (x, y) with any number of edges with x or y as endpoints) are the only graphs to always have a SEFE with any planar graph.…”

Section: Introductionmentioning

confidence: 99%

An SPQR-Tree Approach to Decide Special Cases of Simultaneous Embedding with Fixed Edges

et al. 2009

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Abstract. We present a linear-time algorithm for solving the simultaneous embedding problem with fixed edges (SEFE) for a planar graph and a pseudoforest (a graph with at most one cycle) by reducing it to the following embedding problem: Given a planar graph G, a cycle C of G, and a partitioning of the remaining vertices of G, does there exist a planar embedding in which the induced subgraph on each vertex partite of G \ C is contained entirely inside or outside C? For the latter problem, we present an algorithm that is based on SPQR-trees and has linear running time. We also show how we can employ SPQR-trees to decide SEFE for two planar graphs where one graph has at most two cycles and the intersection is a pseudoforest in linear time. These results give rise to our hope that our SPQR-tree approach might eventually lead to a polynomial-time algorithm for deciding the general SEFE problem for two planar graphs.

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“…These negative results motivated the study of relaxations of simultaneous geometric embeddings. One possibility is to introduce bends along the edges [4,8,9,13], another, to allow that the same vertex occupies different locations in the two drawings [2,3], introducing ambiguity in the mapping.…”

Section: Introductionmentioning

confidence: 99%

Matched Drawings of Planar Graphs

et al. 2008

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Abstract.A natural way to draw two planar graphs whose vertex sets are matched is to assign each matched pair a unique y-coordinate. In this paper we introduce the concept of such matched drawings, which are a relaxation of simultaneous geometric embeddings with mapping. We study which classes of graphs allow matched drawings and show that (i) two 3-connected planar graphs or a 3-connected planar graph and a tree may not be matched drawable, while (ii) two trees or a planar graph and a planar graph of some special families-such as unlabeled level planar (ULP) graphs or the family of "carousel graphs"-are always matched drawable.

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Embedding Graphs Simultaneously with Fixed Edges

Abstract: Abstract. We show that a planar graph and a tree can always be simultaneously embedded with fixed edges and that two outerplanar graphs generally cannot.

Cited by 28 publications

References 9 publications

Intersection Graphs in Simultaneous Embedding with Fixed Edges

Intersection Graphs in Simultaneous Embedding with Fixed Edges

An SPQR-Tree Approach to Decide Special Cases of Simultaneous Embedding with Fixed Edges

Matched Drawings of Planar Graphs

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