974 research outputs found
Visualizing Co-Phylogenetic Reconciliations
We introduce a hybrid metaphor for the visualization of the reconciliations
of co-phylogenetic trees, that are mappings among the nodes of two trees. The
typical application is the visualization of the co-evolution of hosts and
parasites in biology. Our strategy combines a space-filling and a node-link
approach. Differently from traditional methods, it guarantees an unambiguous
and `downward' representation whenever the reconciliation is time-consistent
(i.e., meaningful). We address the problem of the minimization of the number of
crossings in the representation, by giving a characterization of planar
instances and by establishing the complexity of the problem. Finally, we
propose heuristics for computing representations with few crossings.Comment: This paper appears in the Proceedings of the 25th International
Symposium on Graph Drawing and Network Visualization (GD 2017
NodeTrix Planarity Testing with Small Clusters
We study the NodeTrix planarity testing problem for flat clustered graphs
when the maximum size of each cluster is bounded by a constant . We consider
both the case when the sides of the matrices to which the edges are incident
are fixed and the case when they can be chosen arbitrarily. We show that
NodeTrix planarity testing with fixed sides can be solved in
time for every flat clustered graph that can be
reduced to a partial 2-tree by collapsing its clusters into single vertices. In
the general case, NodeTrix planarity testing with fixed sides can be solved in
time for , but it is NP-complete for any . NodeTrix
planarity testing remains NP-complete also in the free sides model when .Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Complexity results for three-dimensional orthogonal graph drawing
AbstractIn this paper we consider the problem of finding three-dimensional orthogonal drawings of maximum degree six graphs from the computational complexity perspective. We introduce a 3SAT reduction framework that can be used to prove the NP-hardness of finding three-dimensional orthogonal drawings with specific constraints. By using the framework we show that, given a three-dimensional orthogonal shape of a graph (a description of the sequence of axis-parallel segments of each edge), finding the coordinates for nodes and bends such that the drawing has no intersection is NP-complete. Conversely, we show that if node coordinates are fixed, finding a shape for the edges that is compatible with a non-intersecting drawing is a feasible problem, which becomes NP-complete if a maximum of two bends per edge is allowed. We comment on the impact of these results on the two open problems of determining whether a graph always admits a drawing with at most two bends per edge and of characterizing orthogonal shapes admitting an orthogonal drawing without intersections
Optimal Morphs of Convex Drawings
We give an algorithm to compute a morph between any two convex drawings of
the same plane graph. The morph preserves the convexity of the drawing at any
time instant and moves each vertex along a piecewise linear curve with linear
complexity. The linear bound is asymptotically optimal in the worst case.Comment: To appear in SoCG 201
A Tipping Point for the Planarity of Small and Medium Sized Graphs
This paper presents an empirical study of the relationship between the
density of small-medium sized random graphs and their planarity. It is well
known that, when the number of vertices tends to infinite, there is a sharp
transition between planarity and non-planarity for edge density d=0.5. However,
this asymptotic property does not clarify what happens for graphs of reduced
size. We show that an unexpectedly sharp transition is also exhibited by small
and medium sized graphs. Also, we show that the same "tipping point" behavior
can be observed for some restrictions or relaxations of planarity (we
considered outerplanarity and near-planarity, respectively).Comment: Appears in the Proceedings of the 28th International Symposium on
Graph Drawing and Network Visualization (GD 2020
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