A star-simple drawing of a graph is a drawing in which adjacent edges do not cross. In contrast, there is no restriction on the number of crossings between two independent edges. We forbid empty lenses, i.e., every lens is required to enclose a vertex, and show that with this restriction $3\cdot(n-4)!$ is an upper bound on the number of crossings between two edges of a star-simple drawing of $K_n$. It follows that $n!$ bounds the total number of crossings in the drawing. This is the first finite upper bound on the number of crossings in star-simple drawings of the complete graph $K_n$ with no empty lens. For a lower bound we construct a star-simple drawing of $K_n$ with no empty lens in which a pair of edges contributes $5^{n/2-2}$ crossings.

We consider the construction of a polygon $P$ with $n$ vertices whose turning angles at the vertices are given by a sequence $A=(\alpha_0,\ldots, \alpha_{n-1})$, $\alpha_i\in (-\pi,\pi)$, for $i\in\{0,\ldots, n-1\}$. The problem of realizing $A$ by a polygon can be seen as that of constructing a straight-line drawing of a graph with prescribed angles at vertices, and hence, it is a special case of the well studied problem of constructing an angle graph. In 2D, we characterize sequences $A$ for which every generic polygon $P\subset \mathbb{R}^2$ realizing $A$ has at least $c$ crossings, for every $c\in \mathbb{N}$, and describe an efficient algorithm that constructs, for a given sequence $A$, a generic polygon $P\subset \mathbb{R}^2$ that realizes $A$ with the minimum number of crossings. In 3D, we describe an efficient algorithm that tests whether a given sequence $A$ can be realized by a (not necessarily generic) polygon $P\subset \mathbb{R}^3$, and for every realizable sequence the algorithm finds a realization.

Graph drawing beyond planarity focuses on drawings of high visual quality for non-planar graphs which are characterized by certain forbidden (edge) crossing configurations. A natural criterion for the quality of a drawing is the number of edge crossings. The question then arises whether beyond-planar drawings have a significantly larger crossing number than unrestricted drawings. Chimani et al. [GD'19] gave bounds for the ratio between the crossing number of three classes of beyond-planar graphs and the unrestricted crossing number. In this paper we extend their results to the main currently known classes of beyond-planar graphs characterized by forbidden edge configurations and answer several of their open questions.

The first algorithm to achieve linear-time performance searching for and identifying a subgraph homeomorphic to $K_{3,3}$ is due to Fellows and Kaschube. Part of the proof of correctness depends on three cases of a straddling bridge of a subgraph homeomorphic to $K_5$ in an input graph. This paper presents a missing fourth case and revises the algorithm with an additional $K_{3,3}$ homeomorph isolator for the condition corresponding to the missing case. This paper also discusses why the prior proof of correctness missed the fourth case and presents a new proof of correctness showing that there are no other missing cases.

The wirelength is one of the key parameters of the quality of embedding graphs into host graphs. To our knowledge, no results for computing the wirelength of embedding irregular graphs into irregular graphs are known in the literature. We develop an algorithm that determines the wirelength of embedding of the Turán graph $T(\ell, 2^p)$, where $2^{n-1} \leq \ell < 2^{n}$ and $1\le p\le \lceil \log_2 \ell\rceil\leq n$, into the incomplete hypercube $I^{\ell}_{n}$. Incomplete hypercubes form an important generalization of hypercubes because they eliminate the restriction on the number of nodes in a system.

The ${\rm T{\small REE}~C{\small ONTAINMENT}}$ problem has many important applications in the study of evolutionary history. Given a phylogenetic network $N$ and a phylogenetic tree $T$ whose leaves are labeled by a set of taxa, it asks if $N$ and $T$ are consistent. While the case of binary $N$ and $T$ has received considerable attention, the more practically relevant variant dealing with biological uncertainty has not. Such uncertainty manifests itself as high-degree vertices ("polytomies") that are "jokers" in the sense that they are compatible with any binary resolution of their children. Contrasting the binary case, we show that this problem, called ${\rm S{\small OFT}~T{\small REE}~C{\small ONTAINMENT}}$, is NP-complete, even if $N$ is a binary, multi-labeled tree in which each taxon occurs at most thrice. On the other hand, we reduce the case that each label occurs at most twice to solving a 2-SAT instance of size $O(|T|^3)$. This implies NP-completeness and polynomial-time solvability on reticulation-visible networks in which the maximum in-degree is bounded by three and two, respectively.

We study drawings of plane quadrangulations such that every inner face realizes a prescribed area. A plane graph is area-universal if for every assignment of non-negative weights to the inner faces, there exists a straight-line drawing such that the area of each inner face equals the weight of the face. It has been conjectured that all plane quadrangulations are area-universal. We develop methods to prove area-universality via reduction to the area-universality of related graphs. This allows us to establish area-universality for large classes of plane quadrangulations. In particular, our methods are strong enough to prove area-universality of all plane quadrangulations with up to 13 vertices.

We find a family of planar bipartite graphs all of whose Lombardi drawings (drawings with circular arcs for edges, meeting at equal angles at the vertices) are nonplanar. We also find families of embedded series-parallel graphs and apex-trees (graphs formed by adding one vertex to a tree) for which there is no planar Lombardi drawing consistent with the given embedding.

In the context of graph layout, many algorithms have been designed to remove node overlapping, and many quality criteria and associated metrics have been proposed to evaluate those algorithms. Unfortunately, a complete comparison of the algorithms based on some metrics that evaluate their quality has never been provided and it is thus difficult for a visualisation designer to select the algorithm that best suits their needs. In this paper, we review 22 metrics available in the literature, classify them according to the quality criteria they try to capture, and select a representative one for each class. Based on the selected metrics,we compare 9 node overlap removal algorithms. Our experiment involves 854 synthetic and real-world graphs. Finally, we propose a JavaScript library containing both the algorithms and the criteria, and we provide a Web platform, AGORA, in which one can upload graphs, apply the algorithms and compare/download the results.

Let $G=(V,E)$ be a vertex-colored graph, where $C$ is the set of colors used to color $V$. The ${\rm G{\small RAPH}~M{\small OTIF}}$ (or $\rm GM$) problem takes as input $G$, a multiset $M$ of colors built from $C$, and asks whether there is a subset $S\subseteq V$ such that (i) $G[S]$ is connected and (ii) the multiset of colors obtained from $S$ equals $M$. The ${\rm C{\small OLORFUL}~G{\small RAPH}~M{\small OTIF}}$ (or ${\rm CGM}$) problem is the special case of $\rm GM$ in which $M$ is a set, and the ${\rm L{\small IST-COLORED}~G{\small RAPH}~M{\small OTIF}}$ (or $\rm LGM$) problem is the extension of $\rm GM$ in which each vertex $v$ of $V$ may choose its color from a list $\mathcal L(v)\subseteq C$ of colors. We study the three problems $\rm GM$, $\rm CGM$, and $\rm LGM$, parameterized by the dual parameter $\ell:=|V|-|M|$. For general graphs, we show that, assuming the strong exponential time hypothesis, $\rm CGM$ has no $(2-\epsilon)^\ell\cdot |V|^{\mathcal O(1)}$-time algorithm, which implies that a previous algorithm, running in $\mathcal O(2^\ell\cdot |E|)$ time is optimal [Betzler et al., IEEE/ACM TCBB 2011]. We also prove that $\rm LGM$ is W[1]-hard with respect to $\ell$ even if we restrict ourselves to lists of at most two colors. Finally, we show that if the input graph is a tree, then $\rm GM$ can be solved in $\mathcal O(3^\ell\cdot |V|)$ time but admits no polynomial-size problem kernel, while $\rm CGM$ can be solved in $\mathcal O(\sqrt{2}^{\ell} + |V|)$ time and admits a polynomial-size problem kernel.