12 research outputs found

### Rectilinear Planarity of Partial 2-Trees

A graph is rectilinear planar if it admits a planar orthogonal drawing
without bends. While testing rectilinear planarity is NP-hard in general (Garg
and Tamassia, 2001), it is a long-standing open problem to establish a tight
upper bound on its complexity for partial 2-trees, i.e., graphs whose
biconnected components are series-parallel. We describe a new O(n^2)-time
algorithm to test rectilinear planarity of partial 2-trees, which improves over
the current best bound of O(n^3 \log n) (Di Giacomo et al., 2022). Moreover,
for partial 2-trees where no two parallel-components in a biconnected component
share a pole, we are able to achieve optimal O(n)-time complexity. Our
algorithms are based on an extensive study and a deeper understanding of the
notion of orthogonal spirality, introduced several years ago (Di Battista et
al, 1998) to describe how much an orthogonal drawing of a subgraph is rolled-up
in an orthogonal drawing of the graph.Comment: arXiv admin note: substantial text overlap with arXiv:2110.00548
Appears in the Proceedings of the 30th International Symposium on Graph
Drawing and Network Visualization (GD 2022

### On the Parameterized Complexity of Bend-Minimum Orthogonal Planarity

Computing planar orthogonal drawings with the minimum number of bends is one
of the most relevant topics in Graph Drawing. The problem is known to be
NP-hard, even when we want to test the existence of a rectilinear planar
drawing, i.e., an orthogonal drawing without bends (Garg and Tamassia, 2001).
From the parameterized complexity perspective, the problem is fixed-parameter
tractable when parameterized by the sum of three parameters: the number of
bends, the number of vertices of degree at most two, and the treewidth of the
input graph (Di Giacomo et al., 2022). We improve this last result by showing
that the problem remains fixed-parameter tractable when parameterized only by
the number of vertices of degree at most two plus the number of bends. As a
consequence, rectilinear planarity testing lies in \FPT~parameterized by the
number of vertices of degree at most two.Comment: Appears in the Proceedings of the 31st International Symposium on
Graph Drawing and Network Visualization (GD 2023

### On the Parameterized Complexity of Computing $st$-Orientations with Few Transitive Edges

Orienting the edges of an undirected graph such that the resulting digraph
satisfies some given constraints is a classical problem in graph theory, with
multiple algorithmic applications. In particular, an $st$-orientation orients
each edge of the input graph such that the resulting digraph is acyclic, and it
contains a single source $s$ and a single sink $t$. Computing an
$st$-orientation of a graph can be done efficiently, and it finds notable
applications in graph algorithms and in particular in graph drawing. On the
other hand, finding an $st$-orientation with at most $k$ transitive edges is
more challenging and it was recently proven to be NP-hard already when $k=0$.
We strengthen this result by showing that the problem remains NP-hard even for
graphs of bounded diameter, and for graphs of bounded vertex degree. These
computational lower bounds naturally raise the question about which structural
parameters can lead to tractable parameterizations of the problem. Our main
result is a fixed-parameter tractable algorithm parameterized by treewidth

### On the Parameterized Complexity of Computing st-Orientations with Few Transitive Edges

Orienting the edges of an undirected graph such that the resulting digraph satisfies some given constraints is a classical problem in graph theory, with multiple algorithmic applications. In particular, an st-orientation orients each edge of the input graph such that the resulting digraph is acyclic, and it contains a single source s and a single sink t. Computing an st-orientation of a graph can be done efficiently, and it finds notable applications in graph algorithms and in particular in graph drawing. On the other hand, finding an st-orientation with at most k transitive edges is more challenging and it was recently proven to be NP-hard already when k = 0. We strengthen this result by showing that the problem remains NP-hard even for graphs of bounded diameter, and for graphs of bounded vertex degree. These computational lower bounds naturally raise the question about which structural parameters can lead to tractable parameterizations of the problem. Our main result is a fixed-parameter tractable algorithm parameterized by treewidth

### Multidimensional Dominance Drawings and Their Applications

In a dominance drawing Γ of a directed acyclic graph (DAG) G, a vertex v is reachable from a vertex u if, and only if all the coordinates of v are greater than or equal to the coordinates of u in Γ. Dominance drawings of DAGs are very important in many areas of research. They combine the aspect of drawing a DAG on the grid with the fact that the transitive closure of the DAG is apparently obvious by the dominance relation between grid points associated with the vertices. The smallest number d for which a given DAG G has a d-dimensional dominance drawing is called dominance drawing dimension, and it is NP-hard to compute. In this paper, we present efficient algorithms for computing dominance drawings of G with a number of dimensions respecting theoretical bounds. We first describe a simple algorithm that shows how to compute a dominance drawing of G from its compressed transitive closure. Next, we describe a more complicated algorithm, which is based on the concept of modular decomposition of G, and obtaining dominance drawings with a lower number of dimensions. Finally, we consider the concept of weak dominance, a relaxed version of the dominance, and we discuss interesting experimental results

### On the Parameterized Complexity of the $s$-Club Cluster Edge Deletion Problem

We study the parameterized complexity of the $s$-Club Cluster Edge Deletion
problem: Given a graph $G$ and two integers $s \ge 2$ and $k \ge 1$, is it
possible to remove at most $k$ edges from $G$ such that each connected
component of the resulting graph has diameter at most $s$? This problem is
known to be NP-hard already when $s = 2$. We prove that it admits a
fixed-parameter tractable algorithm when parameterized by $s$ and the treewidth
of the input graph

### Optimal Orthogonal Drawings of Planar 3-Graphs in Linear Time

This paper addresses a long standing, widely studied, open question: Given a planar 3-graph G (i.e., a planar graph with vertex degree at most three), what is the best computational upper bound to compute a bend-minimum planar orthogonal drawing of G in the variable embedding setting? In this setting the algorithm can choose among the exponentially many planar embeddings of G the one that leads to an orthogonal drawing with the minimum number of bends. We answer the question by describing a linear-time algorithm that computes a bend-minimum planar orthogonal drawing of G. Also, if G is not K4, the drawing has at most one bend per edge. The existence of an orthogonal drawing Ð“ of a planar 3-graph such that Ð“ has the minimum number of bends and at most one bend per edge was previously unknown