Planarity of Streamed Graphs

Lozzo, Giordano Da; Rutter, Ignaz

doi:10.1007/978-3-319-18173-8_11

Cited by 4 publications

(1 citation statement)

References 12 publications

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“…Later on, it was proved [18] that polynomial area could be achieved for trees, tree-maps, and outerplanar graphs if a small number of vertex movements are allowed after each update. The problem has also been studied [13] for general planar graphs, relaxing the requirement that edges have to be straight-line.…”

Section: Introductionmentioning

confidence: 99%

Hierarchical Partial Planarity

Angelini

Bekos

2018

Algorithmica

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In this paper we consider graphs whose edges are associated with a degree of importance, which may depend on the type of connections they represent or on how recently they appeared in the scene, in a streaming setting. The goal is to construct layouts of these graphs in which the readability of an edge is proportional to its importance, that is, more important edges have fewer crossings. We formalize this problem and study the case in which there exist three different degrees of importance. We give a polynomial-time testing algorithm when the graph induced by the two most important sets of edges is biconnected. We also discuss interesting relationships with other constrained-planarity problems.

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Section: Introductionmentioning

confidence: 99%

Hierarchical Partial Planarity

Angelini

Bekos

2018

Algorithmica

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Graph Stories in Small Area

Borrazzo

Lozzo

Frati

et al. 2019

Lecture Notes in Computer Science

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We study the problem of drawing a dynamic graph, where each vertex appears in the graph at a certain time and remains in the graph for a fixed amount of time, called the window size. This defines a graph story, i.e., a sequence of subgraphs, each induced by the vertices that are in the graph at the same time. The drawing of a graph story is a sequence of drawings of such subgraphs. To support readability, we require that each drawing is straight-line and planar and that each vertex maintains its placement in all the drawings. Ideally, the area of the drawing of each subgraph should be a function of the window size, rather than a function of the size of the entire graph, which could be too large. We show that the graph stories of paths and trees can be drawn on a 2W × 2W and on an (8W + 1) × (8W + 1) grid, respectively, where W is the window size. These results are constructive and yield linear-time algorithms. Further, we show that there exist graph stories of planar graphs whose subgraphs cannot be drawn within an area that is only a function of W .

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Planarity of Streamed Graphs

Lozzo

Rutter

2015

Lecture Notes in Computer Science

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Abstract. In this paper we introduce a notion of planarity for graphs that are presented in a streaming fashion. A streamed graph is a stream of edges e1, e2, . . . , em on a vertex set V . A streamed graph is ω-stream planar with respect to a positive integer window size ω if there exists a sequence of planar topological drawings Γi of the graphs Gi = (V, {ej | i ≤ j < i + ω}) such that the common graphis drawn the same in Γi and in Γi+1, for 1 ≤ i < m − ω. The STREAM PLANARITY Problem with window size ω asks whether a given streamed graph is ω-stream planar. We also consider a generalization, where there is an additional backbone graph whose edges have to be present during each time step. These problems are related to several well-studied planarity problems. We show that the STREAM PLANARITY Problem is N P-complete even when the window size is a constant and that the variant with a backbone graph is N P-complete for all ω ≥ 2. On the positive side, we provide O(n + ωm)-time algorithms for (i) the case ω = 1 and (ii) all values of ω provided the backbone graph consists of one 2-connected component plus isolated vertices and no stream edge connects two isolated vertices. Our results improve on the HananiTutte-style O((nm) 3 )-time algorithm proposed by Schaefer [GD'14] for ω = 1.

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Planarity of Streamed Graphs

Cited by 4 publications

References 12 publications

Hierarchical Partial Planarity

Hierarchical Partial Planarity

Graph Stories in Small Area

Planarity of Streamed Graphs

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