Optimal Acyclic Hamiltonian Path Completion for Outerplanar Triangulated st-Digraphs (with Application to Upward Topological Book Embeddings)

On triangulating k-outerplanar graphs

A k-outerplanar graph is a graph that can be drawn in the plane without crossing such that after k-fold removal of the vertices on the outer-face there are no vertices left. In this paper, we study how to triangulate a k-outerplanar graph while keeping its outerplanarity small. Specifically, we show that not all k-outerplanar graphs can be triangulated so that the result is k-outerplanar, but they can be triangulated so that the result is (k+1)-outerplanar.

双连通外平面有向无环图的上向平面性测试解决划分问题

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。

Crossing-Optimal Acyclic HP-Completion for Outerplanar st-Digraphs

Given an embedded planar acyclic digraph G, we define the problem of acyclic hamiltonian path completion with crossing minimization (Acyclic-HPCCM) to be the problem of determining a hamiltonian path completion set of edges such that, when these edges are embedded on G, they create the smallest possible number of edge crossings and turn G to a hamiltonian acyclic digraph. Our results include: 1. We provide a characterization under which a planar st-digraph G is hamiltonian. 2. For an outerplanar st-digraph G, we define the st-polygon decomposition of G and, based on its properties, we develop …

Upward Point Set Embeddability for Convex Point Sets is in $P$

In this paper, we present a polynomial dynamic programming algorithm that tests whether a $n$-vertex directed tree $T$ has an upward planar embedding into a convex point-set $S$ of size $n$. Further, we extend our approach to the class of outerplanar digraphs. This nontrivial and surprising result implies that any given digraph can be efficiently tested for an upward planar embedding into a given convex point set.

Parameterized Algorithms for Upward Planarity

We obtain new parameterized algorithms for the classical problem of determining whether a directed acyclic graph admits an upward planar drawing. Our results include a new fixed-parameter algorithm parameterized by the number of sources, an XP-algorithm parameterized by treewidth, and a fixed-parameter algorithm parameterized by treedepth. All three algorithms are obtained using a novel framework for the problem that combines SPQR tree-decompositions with parameterized techniques. Our approach unifies and pushes beyond previous tractability results for the problem on series-parallel digraphs, …

Planar L-Drawings of Bimodal Graphs

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…