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Register a new userQuasi-upward Planar Drawings with Minimum Curve Complexity
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Added by
Carla Binucci
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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This paper studies the problem of computing quasi-upward planar drawings of bimodal plane digraphs with minimum curve complexity, i.e., drawings such that the maximum number of bends per edge is minimized. We prove that every bimodal plane digraph admits a quasi-upward planar drawing with curve complexity two, which is worst-case optimal. We also show that the problem of minimizing the curve complexity in a quasi-upward planar drawing can be modeled as a min-cost flow problem on a unit-capacity planar flow network. This gives rise to an $tilde{O}(m^frac{4}{3})$-time algorithm that computes a quasi-upward planar drawing with minimum curve complexity; in addition, the drawing has the minimum number of bends when no edge can be bent more than twice. For a contrast, we show bimodal planar digraphs whose bend-minimum quasi-upward planar drawings require linear curve complexity even in the variable embedding setting.
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A fixed-mobile bigraph G is a bipartite graph such that the vertices of one partition set are given with fixed positions in the plane and the mobile vertices of the other part, together with the edges, must be added to the drawing. We assume that G is planar and study the problem of finding, for a given k >= 0, a planar poly-line drawing of G with at most k bends per edge. In the most general case, we show NP-hardness. For k=0 and under additional constraints on the positions of the fixed or mobile vertices, we either prove that the problem is polynomial-time solvable or prove that it belongs to NP. Finally, we present a polynomial-time testing algorithm for a certain type of layered 1-bend drawings.
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We analyze a directed variation of the book embedding problem when the page partition is prespecified and the nodes on the spine must be in topological order (upward book embedding). Given a directed acyclic graph and a partition of its edges into $k$ pages, can we linearly order the vertices such that the drawing is upward (a topological sort) and each page avoids crossings? We prove that the problem is NP-complete for $kge 3$, and for $kge 4$ even in the special case when each page is a matching. By contrast, the problem can be solved in linear time for $k=2$ pages when pages are restricted to matchings. The problem comes from Jack Edmonds (1997), motivated as a generalization of the map folding problem from computational origami.
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We consider straight line drawings of a planar graph $G$ with possible edge crossings. The emph{untangling problem} is to eliminate all edge crossings by moving as few vertices as possible to new positions. Let $fix(G)$ denote the maximum number of vertices that can be left fixed in the worst case. In the emph{allocation problem}, we are given a planar graph $G$ on $n$ vertices together with an $n$-point set $X$ in the plane and have to draw $G$ without edge crossings so that as many vertices as possible are located in $X$. Let $fit(G)$ denote the maximum number of points fitting this purpose in the worst case. As $fix(G)le fit(G)$, we are interested in upper bounds for the latter and lower bounds for the former parameter. For each $epsilon>0$, we construct an infinite sequence of graphs with $fit(G)=O(n^{sigma+epsilon})$, where $sigma<0.99$ is a known graph-theoretic constant, namely the shortness exponent for the class of cubic polyhedral graphs. To the best of our knowledge, this is the first example of graphs with $fit(G)=o(n)$. On the other hand, we prove that $fix(G)gesqrt{n/30}$ for all $G$ with tree-width at most 2. This extends the lower bound obtained by Goaoc et al. [Discrete and Computational Geometry 42:542-569 (2009)] for outerplanar graphs. Our upper bound for $fit(G)$ is based on the fact that the constructed graphs can have only few collinear vertices in any crossing-free drawing. To prove the lower bound for $fix(G)$, we show that graphs of tree-width 2 admit drawings that have large sets of collinear vertices with some additional special properties.
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