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Given a graph G and an integer b , OrthogonalPlanarity is the problem of testing whether G admits a planar orthogonal drawing with at most b bends. OrthogonalPlanarity is known to be NP -complete. We show that this problem belongs to the XP class when parameterized by treewidth. The proof exploits a fixed-parameter tractable approach that uses two more parameters besides treewidth, namely the natural parameter b and the number of vertices with degree at most two of G. Such approach is based on the new concept of sketched orthogonal representation, which synthetically describes a family of equivalent orthogonal drawings. The approach is general enough to be applicable to other related problems, namely HV-Planarity and FlexDraw. Also, in the special case of series-parallel graphs we obtain that both OrthogonalPlanarity and HV-Planarity can be solved in O (n 3 log ⁡ n) time, which improves on the previous O (n 4) bounds for these two. [ABSTRACT FROM AUTHOR]

We study the problem of representing a graph as a storyplan, a recently introduced model for dynamic graph visualization. It is based on a sequence of frames, each showing a subset of vertices and a planar drawing of their induced subgraphs, where vertices appear and disappear over time. Namely, in the StoryPlan problem, we are given a graph and we want to decide whether there exists a total vertex appearance order for which a storyplan exists. We prove that the problem is NP -complete, and complement this hardness with two parameterized algorithms, one in the vertex cover number and one in the feedback edge set number of the input graph. We prove that partial 3-trees always admit a storyplan, which can be computed in linear time. Finally, we show that the problem remains NP -complete if the vertex appearance order is given and we have to choose how to draw the frames. [ABSTRACT FROM AUTHOR]