Scalability considerations for multivariate graph visualization

Real-world, multivariate datasets are frequently too large to show in their entirety on a visual display. Still, there are many techniques we can employ to show useful partial views-sufficient to support incremental exploration of large graph datasets. In this chapter, we first explore the cognitive and architectural limitations which restrict the amount of visual bandwidth available to multivariate graph visualization approaches. These limitations afford several design approaches, which we systematically explore. Finally, we survey systems and studies that exhibit these design strategies to mitigate these perceptual and architectural limitations

On the upward book thickness problem: Combinatorial and complexity results

Among the vast literature concerning graph drawing and graph theory, linear layouts of graphs have been the subject of intense research over the years, both from a combinatorial and from an algorithmic perspective. In particular, upward book embeddings of directed acyclic graphs (DAGs) form a popular class of linear layouts with notable applications, and the upward book thick-ness of a DAG is the minimum number of pages required by any of its upward book embeddings.A long-standing conjecture by Heath, Pemmaraju, and Trenk (1999) states that the upward book thickness of outerplanar DAGs is bounded above by a constant. In this paper, we show that the conjecture holds for subfamilies of upward outerplanar graphs, namely those whose underlying graph is an internally -triangulated outerpath or a cactus, and those whose biconnected components are st-outerplanar graphs.On the complexity side, it is known that deciding whether a graph has upward book thickness k is NP-hard for any fixed k >= 3. We show that the problem, for any k >= 5, remains NP-hard for graphs whose domination number is O(k), but it is fixed-parameter tractable (FPT) in the vertex cover number

On the Upward Book Thickness Problem: Combinatorial and Complexity Results

A long-standing conjecture by Heath, Pemmaraju, and Trenk states that the upward book thickness of outerplanar DAGs is bounded above by a constant. In this paper, we show that the conjecture holds for subfamilies of upward outerplanar graphs, namely those whose underlying graph is an internally-triangulated outerpath or a cactus, and those whose biconnected components are st-outerplanar graphs. On the complexity side, it is known that deciding whether a graph has upward book thickness k is NP-hard for any fixed k≥ 3. We show that the problem, for any k≥ 5, remains NP-hard for graphs whose domination number is O(k), but it is FPT in the vertex cover number

Drawing (Complete) Binary Tanglegrams: Hardness, Approximation, Fixed-Parameter Tractability

A \emph{binary tanglegram} is a pair $$ of binary trees whose leaf sets are in one-to-one correspondence; matching leaves are connected by inter-tree edges. For applications, for example in phylogenetics, it is essential that both trees are drawn without edge crossing and that the inter-tree edges have as few crossings as possible. It is known that finding a drawing with the minimum number of crossings is NP-hard and that the problem is fixed-parameter tractable with respect to that number. We prove that under the Unique Games Conjecture there is no constant-factor approximation for general binary trees. We show that the problem is hard even if both trees are complete binary trees. For this case we give an $O(n^3)$-time 2-approximation and a new and simple fixed-parameter algorithm. We show that the maximization version of the dual problem for general binary trees can be reduced to a version of \textsc{MaxCut} for which the algorithm of Goemans and Williamson yields a 0.878-approximation