Ortho-polygon Visibility Representations of Embedded Graphs

An ortho-polygon visibility representation of an n -vertex embedded graph G (OPVR of G ) is an embedding-preserving drawing of G that maps every vertex to a distinct orthogonal polygon and each edge to a vertical or horizontal visibility between its end-vertices. The vertex complexity of an OPVR of...

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Bibliographic Details
Published in:Algorithmica Vol. 80; no. 8; pp. 2345 - 2383
Main Authors: Di Giacomo, Emilio, Didimo, Walter, Evans, William S., Liotta, Giuseppe, Meijer, Henk, Montecchiani, Fabrizio, Wismath, Stephen K.
Format: Journal Article
Language:English
Published: New York Springer US 01.08.2018
Springer Nature B.V
Subjects:
Algorithm Analysis and Problem Complexity
Algorithms
Complexity
Computer Science
Computer Systems Organization and Communication Networks
Data Structures and Information Theory
Embedding
Graph theory
Graphical representations
Graphs
Mathematics of Computing
Theory of Computation
Visibility
Testing and drawing algorithms
Non-planar graphs
Visibility representations
1-Planarity
Orthogonal polygons
ISSN:0178-4617, 1432-0541
Online Access:Get full text
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Summary:An ortho-polygon visibility representation of an n -vertex embedded graph G (OPVR of G ) is an embedding-preserving drawing of G that maps every vertex to a distinct orthogonal polygon and each edge to a vertical or horizontal visibility between its end-vertices. The vertex complexity of an OPVR of G is the minimum k such that every polygon has at most k reflex corners. We present polynomial time algorithms that test whether G has an OPVR and, if so, compute one of minimum vertex complexity. We argue that the existence and the vertex complexity of an OPVR of G are related to its number of crossings per edge and to its connectivity. More precisely, we prove that if G has at most one crossing per edge (i.e., G is a 1-plane graph), an OPVR of G always exists while this may not be the case if two crossings per edge are allowed. Also, if G is a 3-connected 1-plane graph, we can compute an OPVR of G whose vertex complexity is bounded by a constant in O ( n ) time. However, if G is a 2-connected 1-plane graph, the vertex complexity of any OPVR of G may be Ω ( n ) . In contrast, we describe a family of 2-connected 1-plane graphs for which an embedding that guarantees constant vertex complexity can be computed in O ( n ) time. Finally, we present the results of an experimental study on the vertex complexity of ortho-polygon visibility representations of 1-plane graphs.
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ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-017-0324-2