Contact Representations of Planar Graphs: Extending a Partial Representation is Hard

Chaplick, Steven; Dorbec, Paul; Kratochvíl, Jan; Montassier, Mickael; Stacho, Juraj

doi:10.1007/978-3-319-12340-0_12

Cited by 25 publications

(26 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For weak visibility, the hardness follows easily from results on contact representations by Chaplick et al [8]. For strong visibility, it follows trivially from results by Andreae [1].…”

Section: • Strong Visibility In This Model Two Vertices U V Of G Armentioning

confidence: 65%

“…Every bipartite planar graph is representable as a contact graph of horizontal and vertical segments in the plane [19,31]. Although such representations can be constructed in polynomial time [19,20,41], the problems of extending partial representations of these kinds are NP-hard [8].…”

Section: • Strong Visibility In This Model Two Vertices U V Of G Armentioning

confidence: 99%

“…We also say that s * (t * ) is the left face (right face) of s and t. For example, in Fig. 2, the left face of vertex 5 is s * and the right face of this vertex is the face (s, 7,8,10,14,6,5).…”

Section: Planar St-graphs and Their Propertiesmentioning

confidence: 99%

“…For any rectangular bar visibility representation ψ of G we have: ψ (14) ψ(13) ψ (6) ψ (10) ψ (5) ψ (12) ψ (9) ψ (8) ψ (4) ψ (3) ψ (11) ψ (1) ψ (7) ψ (2) ψ(s) Fig. 5 The graph G, the pertinent digraph of the P-node μ 1 (solid thick edges), the pertinent digraph of the S-node μ 2 (dashed thick edges), the representation ψ(G), the tiling of μ 1 in ψ(G) (solid fill), and the tiling of μ 2 in ψ(G) (patterned fill)…”

Section: In Both Cases B(μ) Is a Rectangle Spanned Between ψ(S μ ) Anmentioning

confidence: 99%

“…Nevertheless, the problem of extending a partial representation of a planar graph to a weak visibility representation is NP-complete [8].…”

Section: Weak Visibilitymentioning

confidence: 99%

See 4 more Smart Citations

The Partial Visibility Representation Extension Problem

et al. 2017

Self Cite

View full text Add to dashboard Cite

(u, v) of G. We study a generalization of the recognition problem where a function ψ defined on a subset V of V (G) is given and the question is whether there is a bar visibility representation ψ of G with ψ(v) = ψ (v) for every v ∈ V . We show that for undirected graphs this problem, and other closely related problems, is NP-complete, but for certain cases involving directed graphs it is solvable in polynomial time.

show abstract

“…For weak visibility, the hardness follows easily from results on contact representations by Chaplick et al [8]. For strong visibility, it follows trivially from results by Andreae [1].…”

Section: • Strong Visibility In This Model Two Vertices U V Of G Armentioning

confidence: 65%

Section: • Strong Visibility In This Model Two Vertices U V Of G Armentioning

confidence: 99%

Section: Planar St-graphs and Their Propertiesmentioning

confidence: 99%

Section: In Both Cases B(μ) Is a Rectangle Spanned Between ψ(S μ ) Anmentioning

confidence: 99%

“…Nevertheless, the problem of extending a partial representation of a planar graph to a weak visibility representation is NP-complete [8].…”

Section: Weak Visibilitymentioning

confidence: 99%

See 3 more Smart Citations

The Partial Visibility Representation Extension Problem

et al. 2017

Self Cite

View full text Add to dashboard Cite

show abstract

Extending partial representations of circle graphs

Chaplick

Fulek

Klavík

2018

Journal of Graph Theory

Self Cite

View full text Add to dashboard Cite

The partial representation extension problem is a recently introduced generalization of the recognition problem. A circle graph is an intersection graph of chords of a circle. We study the partial representation extension problem for circle graphs, where the input consists of a graph G and a partial representation R′ giving some predrawn chords that represent an induced subgraph of G. The question is whether one can extend R ′ to a representation scriptR of the entire graph G, that is, whether one can draw the remaining chords into a partially predrawn representation to obtain a representation of G. Our main result is an scriptO ( n 3 ) time algorithm for partial representation extension of circle graphs, where n is the number of vertices. To show this, we describe the structure of all representations of a circle graph using split decomposition. This can be of independent interest.

show abstract

Extending Upward Planar Graph Drawings

Lozzo

Battista

Frati

2019

Lecture Notes in Computer Science

View full text Add to dashboard Cite

In this paper we study the computational complexity of the Upward Planarity Extension problem, which takes in input an upward planar drawing ΓH of a subgraph H of a directed graph G and asks whether ΓH can be extended to an upward planar drawing of G. Our study fits into the line of research on the extensibility of partial representations, which has recently become a mainstream in Graph Drawing. We show the following results.-First, we prove that the Upward Planarity Extension problem is NP-complete, even if G has a prescribed upward embedding, the vertex set of H coincides with the one of G, and H contains no edge. -Second, we show that the Upward Planarity Extension problem can be solved in O(n log n) time if G is an n-vertex upward planar st-graph. This result improves upon a known O(n 2 )-time algorithm, which however applies to all n-vertex single-source upward planar graphs. -Finally, we show how to solve in polynomial time a surprisingly difficult version of the Upward Planarity Extension problem, in which G is a directed path or cycle with a prescribed upward embedding, H contains no edges, and no two vertices share the same y-coordinate in ΓH .that the drawing of G we seek has to respect a given upward embedding, i.e., a left-to-right order of the edges entering and exiting each vertex. Related problems. Level planar drawings are special upward planar drawings. Klemz and Rote studied the Ordered Level Planarity (OLP) problem [26], where a partial drawing of a level graph is given containing all the vertices and no edges. The problem asks for the existence of a level planar drawing of the graph extending the partial one. They show a tight border of tractability for the problem by proving NP-completeness even if no three vertices have the same y-coordinate and by providing a linear-time algorithm if no two vertices have the same y-coordinate. Brückner and Rutter studied the Partial Level Planarity (PLP) problem [9], that is, the extensibility of a partial drawing of a level graph, which might contain (not necessarily all) vertices and edges. Beside proving NP-completeness even for connected graphs, they provided a quadratic-time algorithm for single-source graphs.The NP-hardness of the Upward Planarity Testing problem [19] directly implies the NP-hardness of the UPE problem, as the former coincides with the special case of the latter in which the partial graph is the empty graph. Further, we have that any instance of the OLP problem in which no λ vertices have the same y-coordinate can be transformed in linear time into an equivalent instance of the UPE problem in which the partial graph contains all the vertices and no edges, and no λ vertices have the same y-coordinate in the partial drawing; moreover, a linear-time reduction can also be performed in the opposite direction. As a consequence of these reductions and of the cited results about the complexity of the OLP problem, we obtain that the UPE problem is NP-hard even if the partial graph contains all the vertices and no edges, and no three vertices s...

show abstract

Contact Representations of Planar Graphs: Extending a Partial Representation is Hard

Abstract: International audienc

Cited by 25 publications

References 22 publications

The Partial Visibility Representation Extension Problem

The Partial Visibility Representation Extension Problem

Extending partial representations of circle graphs

Extending Upward Planar Graph Drawings

Contact Info

Product

Resources

About