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Recognizing Outer 1Planar Graphs in Linear Time
"... A graph is outer 1planar (o1p) if it can be drawn in the plane such that all vertices are on the outer face and each edge is crossed at most once. o1p graphs generalize outerplanar graphs, which can be recognized in linear time and specialize 1planar graphs, whose recognition is N Phard. Our mai ..."
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A graph is outer 1planar (o1p) if it can be drawn in the plane such that all vertices are on the outer face and each edge is crossed at most once. o1p graphs generalize outerplanar graphs, which can be recognized in linear time and specialize 1planar graphs, whose recognition is N Phard. Our main result is a lineartime algorithm that first tests whether a graph G is o1p, and then computes an embedding. Moreover, the algorithm can augment G to a maximal o1p graph. If G is not o1p, then it includes one of six minors (see Fig. 3), which are also detected by the recognition algorithm. Hence, the algorithm returns a positive or negative witness for o1p.
Fixed parameter tractability of crossing minimization of almosttrees
 LECTURE NOTES IN COMPUTER SCIENCE 8242
, 2013
"... We investigate exact crossing minimization for graphs that differ from trees by a small number of additional edges, for several variants of the crossing minimization problem. In particular, we provide fixed parameter tractable algorithms for the 1page book crossing number, the 2page book crossin ..."
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Cited by 2 (1 self)
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We investigate exact crossing minimization for graphs that differ from trees by a small number of additional edges, for several variants of the crossing minimization problem. In particular, we provide fixed parameter tractable algorithms for the 1page book crossing number, the 2page book crossing number, and the minimum number of crossed edges in 1page and 2page book drawings.
A note on 1planar graphs
 EuroCG
"... A graph is 1planar if it can be drawn in the plane such that each of its edges is crossed at most once. We prove a conjecture of Czap and Hudák [6] stating that the edge set of every 1planar graph can be decomposed into a planar graph and a forest. We also provide simple proofs for the following ..."
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A graph is 1planar if it can be drawn in the plane such that each of its edges is crossed at most once. We prove a conjecture of Czap and Hudák [6] stating that the edge set of every 1planar graph can be decomposed into a planar graph and a forest. We also provide simple proofs for the following recent results: (i) an nvertex graph that admits a 1planar drawing with straightline edges has at most 4n − 9 edges [7]; and (ii) every drawing of a maximally dense right angle crossing graph is 1planar [12]. 1
Drawing outer 1planar graphs with few slopes
, 2015
"... A graph is outer 1planar if it admits a drawing where each vertex is on the outer face and each edge is crossed by at most another edge. Outer 1planar graphs are a superclass of the outerplanar graphs and a subclass of the planar partial 3trees. We show that an outer 1planar graph G of bounded d ..."
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A graph is outer 1planar if it admits a drawing where each vertex is on the outer face and each edge is crossed by at most another edge. Outer 1planar graphs are a superclass of the outerplanar graphs and a subclass of the planar partial 3trees. We show that an outer 1planar graph G of bounded degree ∆ admits an outer 1planar straightline drawing that uses O(∆) different slopes, which generalizes a previous result by Knauer et al. about the outerplanar slope number of outerplanar graphs [18]. We also show that O(∆2) slopes suffice to construct a crossingfree straightline drawing of G; the best known upper bound on the planar slope number of planar partial 3trees of bounded degree ∆ is O(∆5) as proved by Jeĺınek et al. [16].
Balanced Circle Packings for Planar Graphs
"... Abstract. In this paper, we study balanced circle packings and circlecontact representations for planar graphs, where the ratio of the largest circle’s diameter to the smallest circle’s diameter is polynomial in the number of circles. We provide a number of positive and negative results for the exi ..."
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Abstract. In this paper, we study balanced circle packings and circlecontact representations for planar graphs, where the ratio of the largest circle’s diameter to the smallest circle’s diameter is polynomial in the number of circles. We provide a number of positive and negative results for the existence of such balanced configurations. 1