Embedded-width: A variant of treewidth for plane graphs

Borradaile, Glencora; Erickson, Jeff; Lê, Hung; Weber, Robbie

doi:10.48550/arxiv.1703.07532

Cited by 1 publication

(2 citation statements)

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“…Therefore, we have cw(δ(G)) ≤ (G)(tw(G) + 2). Finally, the embedded-width emw(G) of G satisfies the relations: emw(G) ≥ and emw(G) ≥ tw(G), by definition [14,26]. Combining the above inequalities, we get the stated bound for the carving-width of δ(G).…”

Section: Lemma 18mentioning

confidence: 94%

“…A tree decomposition of an embedded graph G respects the embedding of G if, for every face f of G, at least one bag contains all the vertices of f [14]. The embedded-width emw(G) of G is the minimum width of any of its tree decompositions that respect the embedding of G. For consistency with other graph-width parameters, in the original definition of this width measure [14] the vertices of the outer face are not required to be in some bag. Here, we adopt the variant presented in [26], where the tree decomposition must also include a bag containing the outer face.…”

Section: Embedded-widthmentioning

confidence: 99%

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C-Planarity Testing of Embedded Clustered Graphs with Bounded Dual Carving-Width

Da Lozzo,

Eppstein,

Goodrich

et al. 2019

Preprint

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For a clustered graph, i.e, a graph whose vertex set is recursively partitioned into clusters, the C-Planarity Testing problem asks whether it is possible to find a planar embedding of the graph and a representation of each cluster as a region homeomorphic to a closed disk such that 1. the subgraph induced by each cluster is drawn in the interior of the corresponding disk, 2. each edge intersects any disk at most once, and 3. the nesting between clusters is reflected by the representation, i.e., child clusters are properly contained in their parent cluster. The computational complexity of this problem, whose study has been central to the theory of graph visualization since its introduction in 1995 [Feng, Cohen, and Eades, Planarity for clustered graphs, ESA'95], has only been recently settled [Fulek and Tóth, Atomic Embeddability, Clustered Planarity, and Thickenability, to appear at SODA'20]. Before such a breakthrough, the complexity question was still unsolved even when the graph has a prescribed planar embedding, i.e, for embedded clustered graphs.We show that the C-Planarity Testing problem admits a single-exponential single-parameter FPT algorithm for embedded clustered graphs, when parameterized by the carving-width of the dual graph of the input. This is the first FPT algorithm for this long-standing open problem with respect to a single notable graph-width parameter. Moreover, in the general case, the polynomial dependency of our FPT algorithm is smaller than the one of the algorithm by Fulek and Tóth. To further strengthen the relevance of this result, we show that the C-Planarity Testing problem retains its computational complexity when parameterized by several other graph-width parameters, which may potentially lead to faster algorithms.

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Section: Lemma 18mentioning

confidence: 94%