Node overlap removal is a necessary step in many scenarios including laying out a graph, or visualizing a tag cloud. Our contribution is a new overlap removal algorithm that iteratively builds a Minimum Spanning Tree on a Delaunay triangulation of the node centers and removes the node overlaps by "growing" the tree. The algorithm is simple to implement, yet it produces high quality layouts. According to our experiments it runs several times faster than the current state-of-the-art methods.

Given an undirected graph, one can assign directions to each of the edges of the graph, thus orienting the graph. To be as egalitarian as possible, one may wish to find an orientation such that no vertex is unfairly hit with too many arcs directed into it. We discuss how this objective arises in problems resulting from telecommunications. We give optimal, polynomial-time algorithms for: finding an orientation that minimizes the lexicographic order of the indegrees and finding a strongly-connected orientation that minimizes the maximum indegree. We show that minimizing the lexicographic order of the indegrees is NP-hard when the resulting orientation is required to be acyclic.

A cycle basis in an undirected graph is a minimal set of simple cycles whose symmetric differences include all Eulerian subgraphs of the given graph. We define a rooted cycle basis to be a cycle basis in which all cycles contain a specified root edge, and we investigate the algorithmic problem of constructing rooted cycle bases. We show that a given graph has a rooted cycle basis if and only if the root edge belongs to its 2-core and the 2-core is 2-vertex-connected, and that constructing such a basis can be performed efficiently. We show that in an unweighted or positively weighted graph, it is possible to find the minimum weight rooted cycle basis in polynomial time. Additionally, we show that it is NP-complete to find a fundamental rooted cycle basis (a rooted cycle basis in which each cycle is formed by combining paths in a fixed spanning tree with a single additional edge) but that the problem can be solved by a fixed-parameter-tractable algorithm when parameterized by clique-width.

Recently, a new way of avoiding crossings in straight-line drawings of non-planar graphs has been introduced. The idea of partial edge drawings (PED) is to drop the middle part of edges and rely on the remaining edge parts called stubs. We focus on symmetric partial edge drawings (SPEDs) that require the two stubs of an edge to be of equal length. In this way, the stub at the other endpoint of an edge assures the viewer of the edge's existence. We also consider an additional homogeneity constraint that forces the stub lengths to be a given fraction $\delta$ of the edge lengths ($\delta$-SHPED). Given length and direction of a stub, this model helps to infer the position of the opposite stub. We show that, for a fixed stub–edge length ratio $\delta$, not all graphs have a $\delta$-SHPED. Specifically, we show that $K_{165}$ does not have a $1/4$-SHPED, while bandwidth-$k$ graphs always have a $\Theta(1/\sqrt{k})$-SHPED. We also give bounds for complete bipartite graphs. Further, we consider the problem ${\rm M{\small AX} SPED}$ where the task is to compute the SPED of maximum total stub length that a given straight-line drawing contains. We present an efficient solution for 2-planar drawings and a 2-approximation algorithm for the dual problem of minimizing the total amount of erased ink.

An octilinear drawing of a planar graph is one in which each edge is drawn as a sequence of horizontal, vertical and diagonal at $45^\circ$ and $-45^\circ$ line-segments. For such drawings to be readable, special care is needed in order to keep the number of bends small. Since the problem of finding planar octilinear drawings with minimum number of bends is NP-hard, in this paper we focus on upper and lower bounds. From a recent result of Keszegh et al. on the slope number of planar graphs, we can derive an upper bound of $4n-10$ bends for planar graphs with $n$ vertices and maximum degree $8$. We considerably improve this general bound and corresponding previous ones for triconnected planar graphs of maximum degree $4$, $5$ and $6$. We also derive non-trivial lower bounds for these three classes of graphs by a technique inspired by the network flow formulation of Tamassia for computing bend optimal orthogonal drawings.

This is the first of two chapters of a work in which we consider the unrestricted, minimal, and bounded representation problems for unit interval (UIG) and unit circular-arc (UCA) graphs. In the unrestricted version, a proper circular-arc (PCA) model ${\mathcal{M}}$ is given and the goal is to obtain an equivalent UCA model ${\mathcal{U}}$. In the bounded version, ${\mathcal{M}}$ is given together with some lower and upper bounds that the beginning points of ${\mathcal{U}}$ must satisfy. In the minimal version, we have to find a minimal model equivalent to ${\mathcal{M}}$, in which the circumference of the circle and length of the arcs must be simultaneously as small as possible. In this chapter we motivate these problems from an historical perspective, and we develop the theoretical framework required for the algorithms in Chapter II. We present new characterizations of those PCA models that have equivalent UCA models, and of those UCA models with a circle of circumference ${c}$ and the arcs of length $\ell$. We also prove that every UCA model is equivalent to a minimal one. We remark that all our results are of an algorithmic nature and can be readily employed to solve the problems at hand, even though these algorithms are not as efficient as those in Chapter II.

In a fan-planar drawing of a graph there is no edge that crosses two other independent edges. We study 2-layer fan-planar drawings, i.e., fan-planar drawings such that the vertices are restricted to two distinct horizontal layers and edges are straight-line segments that connect vertices of different layers. We characterize 2-layer fan-planar drawable graphs and describe a linear-time testing and embedding algorithm for biconnected graphs. We also study the relationship between 2-layer fan-planar graphs and 2-layer right-angle crossing graphs.

${\rm H{\small ITTING}~S{\small ET}}$ is a classic problem in combinatorial optimization. Its input consists of a set system $\mathcal F$ over a finite universe $U$ and an integer $t$; the question is whether there is a set of $t$ elements that intersects every set in $\mathcal F$. The ${\rm H{\small ITTING}~S{\small ET}}$ problem parameterized by the size of the solution is a well-known W[2]-complete problem in parameterized complexity theory. In this paper we investigate the complexity of ${\rm H{\small ITTING}~S{\small ET}}$ under various structural parameterizations of the input. Our starting point is the folklore result that ${\rm H{\small ITTING}~S{\small ET}}$ is polynomial-time solvable if there is a tree $T$ on vertex set $U$ such that the sets in $\mathcal F$ induce connected subtrees of $T$. We consider the case that there is a treelike graph with vertex set $U$ such that the sets in $\mathcal F$ induce connected subgraphs; the parameter of the problem is a measure of how treelike the graph is. Our main positive result is an algorithm that, given a graph $G$ with cyclomatic number $k$, a collection $\mathcal P$ of simple paths in $G$, and an integer $t$, determines in time $2^{5k} (|G| +|\mathcal P|)^{\mathcal O(1)}$ whether there is a vertex set of size $t$ that hits all paths in $\mathcal P$. It is based on a connection to the 2-SAT problem in multiple valued logic. For other parameterizations we derive W[1]-hardness and para-NP-completeness results.

Given a complete directed graph G with weights on the vertices and on the arcs, a θ-improper k-coloring is an assignment of at most k different colors to the vertices of G such that the weight of every vertex v is greater, by a given factor 1/θ, than the sum of the weights on the arcs (u,v) entering v with the tail u of the same color as v. For a given real number θ and an integer k, the Partial Directed Weigthed Improper Coloring Problem (PDWICP) is to determine the order of the largest induced subgraph G′ of G such that G′ admits a θ-improper k-coloring. This problem is motivated by a practical channel assignment application where the objective is to maximize the number of simultaneously communicating mobiles in a wireless network. We consider three constructive algorithms for the standard vertex coloring problem, and adapt them to the PDWICP. We show that they perform better than today's phone operator systems based on decentralized channel assignment strategies such as fractional frequency reuse.

Within the field of phylogenetics there is growing interest in measures for summarising the dissimilarity, or incongruence, of two or more phylogenetic trees. Many of these measures are NP-hard to compute and this has stimulated a considerable volume of research into fixed parameter tractable algorithms. In this article we use Monadic Second Order logic to give alternative, compact proofs of fixed parameter tractability for several well-known incongruence measures. In doing so we wish to demonstrate the considerable potential of MSOL - machinery still largely unknown outside the algorithmic graph theory community - within phylogenetics. A crucial component of this work is the observation that many measures, when bounded, imply the existence of an agreement forest of bounded size, which in turn implies that an auxiliary graph structure, the display graph, has bounded treewidth. It is this bound on treewidth that makes the machinery of MSOL available for proving fixed parameter tractability.