2013
DOI: 10.1007/978-3-319-03841-4_8
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Straight-Line Grid Drawings of 3-Connected 1-Planar Graphs

Abstract: Abstract.A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. In general, 1-planar graphs do not admit straightline drawings. We show that every 3-connected 1-planar graph has a straight-line drawing on an integer grid of quadratic size, with the exception of a single edge on the outer face that has one bend. The drawing can be computed in linear time from any given 1-planar embedding of the graph.

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Cited by 48 publications
(73 citation statements)
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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%
See 1 more Smart Citation
“…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%
“…Particular attention has been given to recognition and complexity problems (see, e.g., [4,8,17,27]), straight-line drawings (see, e.g., [2,39]), right-angle crossing drawings (see, e.g., [15,18]), and visibility representations (see, e.g, [6,7,19]); see also [26] for additional references and topics. In addition, two recent papers [20,29] study visibility representations of non-planar graphs where the edges are horizontal and vertical lines of sight and each vertex consists of two segments sharing an end-point.…”
Section: Introductionmentioning
confidence: 99%
“…Hence, the only blocks that are between B x 1 and B y 1 on the spine are descendant-blocks of B x 1 on the blocktree. Combining the above restrictions, we distinguish three cases for B x 2 : (c.1) B x 2 = B x 1 and x 2 appears after x 1 , (c.2) B x 2 is a descendant of B x 1 , and (c.3) B x 2 = B y 1 and x 2 appears before y 1 In the first and third case, edges e 1 and e 2 are bridging 2-hops with one endpoint on the same block. In the second case, again by definition, B y 2 is also a descendant-block of B x 1 (since B x 2 and B y 2 have the same parent-block).…”
Section: The Two-level Casementioning
confidence: 99%
“…A 1-planar topological graph is called planar-maximal or simply maximal, if the addition of a non-crossed edge is not possible. The following lemma, proven in many earlier papers, shows that two crossing edges induce a K 4 , as the missing edges can be added without introducing new crossings; see, e.g., [1]. …”
mentioning
confidence: 92%
“…Again, Algorithm 1 can check these two conditions and augment the graph for a P-node in time O (1), which results in a running time of O(n) for ll. 4 -12.…”
Section: Lemma 4 Let U V Be the Vertices In The Skeleton Of A P-nodmentioning
confidence: 99%