Automatic graph drawing and readability of diagrams
IEEE Transactions on Systems, Man and Cybernetics
Alpha-algorithms for incremental planarity testing (preliminary version)
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
On-line planar graph embedding
Journal of Algorithms
SIAM Journal on Computing
Coloring precolored perfect graphs
Journal of Graph Theory
A Linear Time Algorithm for Embedding Graphs in an Arbitrary Surface
SIAM Journal on Discrete Mathematics
Journal of the ACM (JACM)
Computing Orthogonal Drawings with the Minimum Number of Bends
IEEE Transactions on Computers
Constraints in Graph Drawing Algorithms
Constraints
Fast Incremental Planarity Testing
ICALP '92 Proceedings of the 19th International Colloquium on Automata, Languages and Programming
A Linear Time Implementation of SPQR-Trees
GD '00 Proceedings of the 8th International Symposium on Graph Drawing
Short path queries in planar graphs in constant time
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
2-Restricted extensions of partial embeddings of graphs
European Journal of Combinatorics - Special issue: Topological graph theory II
NP completeness of the edge precoloring extension problem on bipartite graphs
Journal of Graph Theory
Graph-Theoretic Concepts in Computer Science
Embedding graphs simultaneously with fixed edges
GD'06 Proceedings of the 14th international conference on Graph drawing
The SPQR-tree data structure in graph drawing
ICALP'03 Proceedings of the 30th international conference on Automata, languages and programming
Simultaneous geometric graph embeddings
GD'07 Proceedings of the 15th international conference on Graph drawing
Simultaneous graph embeddings with fixed edges
WG'06 Proceedings of the 32nd international conference on Graph-Theoretic Concepts in Computer Science
Simultaneous embedding of planar graphs with few bends
GD'04 Proceedings of the 12th international conference on Graph Drawing
IWOCA'10 Proceedings of the 21st international conference on Combinatorial algorithms
A kuratowski-type theorem for planarity of partially embedded graphs
Proceedings of the twenty-seventh annual symposium on Computational geometry
Extending partial representations of interval graphs
TAMC'11 Proceedings of the 8th annual conference on Theory and applications of models of computation
Simultaneous embedding of embedded planar graphs
ISAAC'11 Proceedings of the 22nd international conference on Algorithms and Computation
Journal of Discrete Algorithms
Extending partial representations of function graphs and permutation graphs
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
A Kuratowski-type theorem for planarity of partially embedded graphs
Computational Geometry: Theory and Applications
Disconnectivity and relative positions in simultaneous embeddings
GD'12 Proceedings of the 20th international conference on Graph Drawing
Toward a theory of planarity: hanani-tutte and planarity variants
GD'12 Proceedings of the 20th international conference on Graph Drawing
Topological morphing of planar graphs
Theoretical Computer Science
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We study the following problem: Given a planar graph G and a planar drawing (embedding) of a subgraph of G, can such a drawing be extended to a planar drawing of the entire graph G? This problem fits the paradigm of extending a partial solution to a complete one, which has been studied before in many different settings. Unlike many cases, in which the presence of a partial solution in the input makes hard an otherwise easy problem, we show that the planarity question remains polynomial-time solvable. Our algorithm is based on several combinatorial lemmata which show that the planarity of partially embedded graphs meets the "on-cas" behaviour -- obvious necessary conditions for planarity are also sufficient. These conditions are expressed in terms of the interplay between (a) rotation schemes and containment relationships between cycles and (b) the decomposition of a graph into its connected, biconnected, and triconnected components. This implies that no dynamic programming is needed for a decision algorithm and that the elements of the decomposition can be processed independently. Further, by equipping the components of the decomposition with suitable data structures and by carefully splitting the problem into simpler subproblems, we improve our algorithm to reach linear-time complexity. Finally, we consider several generalizations of the problem, e.g. minimizing the number of edges of the partial embedding that need to be rerouted to extend it, and argue that they are NP-hard. Also, we show how our algorithm can be applied to solve related Graph Drawing problems.