Re-embeddings of Maximum 1-Planar Graphs

Suzuki, Yusuke

doi:10.1137/090746835

Cited by 67 publications

(81 citation statements)

References 6 publications

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“…We call the four edges of the K 4 different from (u, v) and (w, z) cycle edges of (u, v) and (w, z) -they form a 4-cycle. Note that a 1-plane graph can always be made crossingaugmented in O(n) time, by adding the missing cycle edges without introducing any new edge crossings (see, e.g., [2,35]). …”

Section: Types Of Crossings In 1-plane Graphsmentioning

confidence: 99%

“…The linear time complexity is a consequence of the fact that a 1-plane graph has O(n) edges [35] and that Schnyder trees can be constructed in O(n) time [32].…”

Section: Claimmentioning

confidence: 99%

“…A drawing of H is obtained by applying the compaction step of the TSM framework. Since the number of bends of H is O(n) and since the number of crossings of a 1-plane graph is O(n) (see, e.g., [35]), this step is executed in O(n) time and produces a drawing on an integer grid of size O(n) × O(n).…”

Section: Claimmentioning

confidence: 99%

“…It remains to extend the argument of the proof to any 1-planar embedding of G. To this aim, we observe that graph K together with the edge (u, v) has a unique 1-planar embedding [35]. This, together with the fact that no two copies of K can intersect one another without violating 1-planarity, implies that G has a unique embedding up to a renaming of the n + 1 copies of K, except for the edge (u, v).…”

Section: Vertex Complexity Bounds For 2-connected 1-plane Graphsmentioning

confidence: 99%

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Ortho-polygon Visibility Representations of Embedded Graphs

et al. 2017

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An ortho-polygon visibility representation of an n-vertex embedded graph G (OPVR of G) is an embedding-preserving drawing of G that maps every vertex to a distinct orthogonal polygon and each edge to a vertical or horizontal visibility between its end-vertices. The vertex complexity of an OPVR of G is the minimum k such that every polygon has at most k reflex corners. We present polynomial time algorithms that test whether G has an OPVR and, if so, compute one of minimum vertex complexity. We argue that the existence and the vertex complexity of an OPVR of G are related to its number of crossings per edge and to its connectivity. More precisely, we prove that if G has at most one crossing per edge (i.e., G is a 1-plane graph), an OPVR of G always exists while this may not be the case if two crossings per edge are allowed. Also, if G is a 3-connected 1-plane graph, we can compute an OPVR of G whose vertex complexity is bounded by a constant in O(n) time. However, if G is a 2-connected 1-plane graph, the vertex complexity of any OPVR of G may be Ω(n). In contrast, we describe a family of 2-connected 1-plane graphs for which an embedding that guarantees constant vertex complexity can be computed in O(n) time. Finally, we present the results of an experimental study on the vertex complexity of ortho-polygon visibility representations of 1-plane graphs.

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Section: Types Of Crossings In 1-plane Graphsmentioning

confidence: 99%

“…The linear time complexity is a consequence of the fact that a 1-plane graph has O(n) edges [35] and that Schnyder trees can be constructed in O(n) time [32].…”

Section: Claimmentioning

confidence: 99%

Section: Claimmentioning

confidence: 99%

Section: Vertex Complexity Bounds For 2-connected 1-plane Graphsmentioning

confidence: 99%

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Ortho-polygon Visibility Representations of Embedded Graphs

et al. 2017

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“…The 2-quasi-planar graphs are nothing but the planar graphs. References for these definitions are [28,34,41,43]. It is clear that every p-planar graph is…”

mentioning

confidence: 99%

On quasi-planar graphs: Clique-width and logical description

Courcelle

2020

Discrete Applied Mathematics

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Motivated by the construction of FPT graph algorithms parameterized by clique-width or tree-width, we study graph classes for which treewidth and clique-width are linearly related. This is the case for all graph classes of bounded expansion, but in view of concrete applications, we want to have "small" constants in the comparisons between these width parameters.We focus our attention on graphs that can be drawn in the plane with limited edge crossings, for an example, at most p crossings for each edge. These graphs are called p-planar. We consider a more general situation where the graph of edge crossings must belong to a fixed class of graphs D. For p-planar graphs, D is the class of graphs of degree at most p. We prove that tree-width and clique-width are linearly related for graphs drawable with a graph of crossings of bounded average degree.We prove that the class of 1-planar graphs, although conceptually close to that of planar graphs, is not characterized by a monadic second-order sentence. We identify two subclasses that are.

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Cyclic 4‐Colorings of Graphs on Surfaces

Nakamoto

Noguchi

Ozeki

2015

Journal of Graph Theory

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To attack the Four Color Problem, in 1880, Tait gave a necessary and sufficient condition for plane triangulations to have a proper 4‐vertex‐coloring: a plane triangulation G has a proper 4‐vertex‐coloring if and only if the dual of G has a proper 3‐edge‐coloring. A cyclic coloring of a map G on a surface F2 is a vertex‐coloring of G such that any two vertices x and y receive different colors if x and y are incident with a common face of G. In this article, we extend the result by Tait to two directions, that is, considering maps on a nonspherical surface and cyclic 4‐colorings.

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Re-embeddings of Maximum 1-Planar Graphs

Cited by 67 publications

References 6 publications

Ortho-polygon Visibility Representations of Embedded Graphs

Ortho-polygon Visibility Representations of Embedded Graphs

On quasi-planar graphs: Clique-width and logical description

Cyclic 4‐Colorings of Graphs on Surfaces

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