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We study the complexity of testing whether a biconnected graph G = (V , E) is planar with the constraint that some cyclic orders of the edges incident to its vertices are allowed while some others are forbidden. The allowed cyclic orders are described by associating every vertex v of G with a set D (v) of FPQ-trees. Let tw be the treewidth of G and let D max be the maximum number of FPQ-trees per vertex. We show that the problem is FPT when parameterized by tw + D max , paraNP-hard when parameterized by D max , and W[1]-hard when parameterized by tw. We also consider NodeTrix planar representations of clustered graphs, where clusters are adjacency matrices and inter-cluster edges are non-intersecting simple curves. We prove that NodeTrix planarity with fixed sides is FPT when parameterized by the size of clusters plus the treewidth of the graph obtained by collapsing clusters to single vertices, provided that this graph is biconnected. [ABSTRACT FROM AUTHOR]

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]

Abstract An HV-graph is a planar graph with vertex-degree at most four such that each edge is labeled either H (horizontal) or V (vertical). The HV-planarity testing problem asks whether an HV-graph admits an HV-drawing , that is, a planar drawing such that each edge with label H is drawn as a horizontal segment and each edge with label V is drawn as a vertical segment. We prove that the HV-planarity testing problem is NP-complete even for graphs with vertex-degree at most three, which answers an open question posed by both Manuch et al. [30] and Durocher et al. [17]. On the positive side, we prove that the HV-planarity testing problem can be solved in polynomial-time for series-parallel graphs. This result significantly extends a previous result by Durocher et al. about HV-planarity testing of biconnected outerplanar graphs of maximum vertex-degree three. Highlights • HV-planarity testing is NP-complete, even for graphs with vertex-degree at most 3. • Polynomial-time HV-planarity testing for 2-connected series-parallel graphs. • Polynomial-time HV-planarity testing for partial 2-trees. [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]

Abstract: We introduce and study -EMST drawings, i.e., planar straight-line drawings of trees such that, for any fixed , the distance between any two vertices is at least the length of the longest edge in the path connecting them. -EMST drawings are good approximations of Euclidean minimum spanning trees. While it is known that only trees with bounded degree have a Euclidean minimum spanning tree realization, we show that every tree T has a -EMST drawing for any given . We also present drawing algorithms that compute -EMST drawings of trees with bounded degree in polynomial area. As a byproduct of one of our techniques, we improve the best known area upper bound for Euclidean minimum spanning tree realizations of complete binary trees. [Copyright &y& Elsevier]

A simple topological graph is k -quasiplanar (k ≥ 2) if it contains no k pairwise crossing edges, and k -planar if no edge is crossed more than k times. In this paper, we explore the relationship between k -planarity and k -quasiplanarity to show that, for k ≥ 2 , every k -planar simple topological graph can be transformed into a (k + 1) -quasiplanar simple topological graph. [ABSTRACT FROM AUTHOR]

A drawing of a graph such that the vertices are drawn as points along a line and each edge is a circular arc in one of the two half-planes defined by this line is called a 2 -page drawing. If all edges are in the same half-plane, the drawing is called a 1 -page drawing. We want to compute 1 -page and 2 -page drawings of planar graphs such that the number of crossings per edge does not depend on the number of vertices. We show that for any constant k , there exist planar graphs that require more than k crossings per edge in both 1 -page and 2 -page drawings. We then prove that if the vertex degree is bounded by Δ , every planar 3-tree has a 2 -page drawing with a number of crossings per edge that only depends on Δ . Finally, we show a similar result for 1 -page drawings of partial 2 -trees. [ABSTRACT FROM AUTHOR]