Fabrizio Frati
We show how to test in linear time whether an outerplanar graph admits a planar rectilinear drawing, both if the graph has a prescribed plane embedding that the drawing has to respect and if it does not. Our algorithm returns a planar rectilinear drawing if the graph admits one.
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Internally-Convex Drawings of Outerplanar Graphs in Small Area
研究外平面图在小面积内的内凸绘制问题，提出算法在O(n^1.5)面积内计算相关绘制，还给出内面为严格凸多边形时不同类型外平面图的面积界，如弱对偶为路径且最大面大小为k的为Θ(nk²)，直径受d限制的为Θ(nd²)。
In this paper, we study planar drawings of maximal outerplanar graphs with the objective of achieving small height. A recent paper gave an algorithm for such drawings that is within a factor of 4 of the optimum height. In this paper, we substantially improve the approximation factor to become 2. The main ingredient is to define a new parameter of outerplanar graphs (the so-called umbrella depth, obtained by recursively splitting the graph into graphs called umbrellas). We argue that the height of any poly-line drawing must be at least the umbrella depth, and then devise an algorithm that achie…
In this paper we study a family of algorithms, introduced by Chan [SODA 1999] and called LR-algorithms, for drawing ordered rooted binary trees. In particular, we are interested in constructing LR-drawings (that are drawings obtained via LR-algorithms) with small width. Chan showed three different LR-algorithms that achieve, for an ordered rooted binary tree with $n$ nodes, width $O(n^{0.695})$, width $O(n^{0.5})$, and width $O(n^{0.48})$. We prove that, for every $n$-node ordered rooted binary tree, an LR-drawing with minimum width can be constructed in $O(n^{1.48})$ time. Further, we show an…
In a planar L-drawing of a directed graph (digraph) each edge e is represented as a polyline composed of a vertical segment starting at the tail of e and a horizontal segment ending at the head of e. Distinct edges may overlap, but not cross. Our main focus is on bimodal graphs, i.e., digraphs admitting a planar embedding in which the incoming and outgoing edges around each vertex are contiguous. We show that every plane bimodal graph without 2-cycles admits a planar L-drawing. This includes the class of upward-plane graphs. Finally, outerplanar digraphs admit a planar L-drawing - although the…
Upward Planarity Testing of Biconnected Outerplanar DAGs Solves Partition
本文通过$O(n)$时间归约证明，双连通外平面DAG的上向平面性测试等价于划分问题，并给出反向结果，为高效算法存在性设立首个障碍。
It is proven that every set $S$ of distinct points in the plane with cardinality $\lceil \frac{\sqrt{\log_2 n}-1}{4} \rceil$ can be a subset of the vertices of a crossing-free straight-line drawing of any planar graph with $n$ vertices. It is also proven that if $S$ is restricted to be a one-sided convex point set, its cardinality increases to $\lceil \sqrt[3]{n} \rceil$. The proofs are constructive and give rise to O(n)-time drawing algorithms. As a part of our proofs, we show that every maximal planar graph contains a large induced biconnected outerplanar graphs and a large induced outerpath…
A plane graph is rectilinear planar if it admits an embedding-preserving straight-line drawing where each edge is either horizontal or vertical. We prove that rectilinear planarity testing can be solved in optimal $O(n)$ time for any plane series-parallel graph $G$ with $n$ vertices. If $G$ is rectilinear planar, an embedding-preserving rectilinear planar drawing of $G$ can be constructed in $O(n)$ time. Our result is based on a characterization of rectilinear planar series-parallel graphs in terms of intervals of orthogonal spirality that their components can have, and it leads to an algorith…
Given a connected outerplanar graph G of pathwidth p, we give an algorithm to add edges to G to get a supergraph of G, which is 2-vertex-connected, outerplanar and of pathwidth O(p). This settles an open problem raised by Biedl, in the context of computing minimum height planar straight line drawings of outerplanar graphs, with their vertices placed on a two dimensional grid. In conjunction with the result of this paper, the constant factor approximation algorithm for this problem obtained by Biedl for 2-vertex-connected outerplanar graphs will work for all outer planar graphs.
Rectilinear Planarity of Partial 2-Trees
本文提出O(n^2)时间算法测试部分2-树的直线平面性，改进现有O(n^3 log n)界限，并在特定条件下达到最优O(n)时间，基于对正交螺旋性的深入研究。
Given two planar graphs that are defined on the same set of vertices, a RAC simultaneous drawing is one in which each graph is drawn planar, there are no edge overlaps and the crossings between the two graphs form right angles. The geometric version restricts the problem to straight-line drawings. It is known, however, that there exists a wheel and a matching which do not admit a geometric RAC simultaneous drawing. In order to enlarge the class of graphs that admit RAC simultaneous drawings, we allow bends in the resulting drawings. We prove that two planar graphs always admit a RAC simultaneo…
In streamed graph drawing, a planar graph, G, is given incrementally as a data stream and a straight-line drawing of G must be updated after each new edge is released. To preserve the mental map, changes to the drawing should be minimized after each update, and Binucci et al.show that exponential area is necessary and sufficient for a number of streamed graph drawings for trees if edges are not allowed to move at all. We show that a number of streamed graph drawings can, in fact, be done with polynomial area, including planar streamed graph drawings of trees, tree-maps, and outerplanar graphs,…
Thomassen characterized some 1-plane embedding as the forbidden configuration such that a given 1-plane embedding of a graph is drawable in straight-lines if and only if it does not contain the configuration [C. Thomassen, Rectilinear drawings of graphs, J. Graph Theory, 10(3), 335-341, 1988]. In this paper, we characterize some 1-plane embedding as the forbidden configuration such that a given 1-plane embedding of a graph can be re-embedded into a straight-line drawable 1-plane embedding of the same graph if and only if it does not contain the configuration. Re-embedding of a 1-plane embeddin…
Beyond Degree Four: Near-Orthogonal Planar Drawings
研究最大度数无限制的图的平面折线图绘制，聚焦接近正交的绘制。证明测试至多\(h\)个非正交面的绘制问题是NP完全的，在固定和可变嵌入设置下，分别给出基于外平面性指数、\(h\)、树宽等参数的FPT算法及多项式时间近似方案。
Quantum Graph Drawing
本文研究2-层绘制与书式布局中的量子算法，基于Grover搜索给出相对经典精确算法至少二次加速的框架，并通过D-Wave实验表明量子退火在2-层交叉最小化上具有竞争力。
In an upward-planar L-drawing of a directed acyclic graph (DAG) each edge $e$ is represented as a polyline composed of a vertical segment with its lowest endpoint at the tail of $e$ and of a horizontal segment ending at the head of $e$. Distinct edges may overlap, but not cross. Recently, upward-planar L-drawings have been studied for $st$-graphs, i.e., planar DAGs with a single source $s$ and a single sink $t$ containing an edge directed from $s$ to $t$. It is known that a plane $st$-graph, i.e., an embedded $st$-graph in which the edge $(s,t)$ is incident to the outer face, admits an upward-…
Weakly Leveled Planarity with Bounded Span
本文研究有界跨度的弱分层平面图绘制问题，证明其关于跨度参数 para-NP 难，并给出关于顶点覆盖数和树深的参数化复杂性结果，同时证明循环树等图类具有对数级或常数级跨度上界。
The Parameterized Complexity of Problems on Outer k-Planar Graphs
本文研究外k-平面图上图问题的参数化复杂性，证明Binary CSP等少数问题仍难解，多数问题固定参数可处理，还建立了外k-平面图的多项结构结果并明确其在图参数层次中的位置。
A k-queue layout is a special type of a linear layout, in which the linear order avoids (k+1)-rainbows, i.e., k+1 independent edges that pairwise form a nested pair. The optimization goal is to determine the queue number of a graph, i.e., the minimum value of k for which a k-queue layout is feasible. Recently, Dujmović et al. [J. ACM, 67(4), 22:1-38, 2020] showed that the queue number of planar graphs is at most 49, thus settling in the positive a long-standing conjecture by Heath, Leighton and Rosenberg. To achieve this breakthrough result, their approach involves three different techniques: …
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