Complexity and algorithms for reasoning about time: a graph-theoretic approach
Journal of the ACM (JACM)
Graph classes: a survey
Maintaining knowledge about temporal intervals
Communications of the ACM
Algorithms and Complexity of Sandwich Problems in Graphs (Extended Abstract)
WG '93 Proceedings of the 19th International Workshop on Graph-Theoretic Concepts in Computer Science
Reasoning about temporal relations: The tractable subalgebras of Allen's interval algebra
Journal of the ACM (JACM)
Algorithmic Graph Theory and Perfect Graphs (Annals of Discrete Mathematics, Vol 57)
Algorithmic Graph Theory and Perfect Graphs (Annals of Discrete Mathematics, Vol 57)
A simple 3-sweep LBFS algorithm for the recognition of unit interval graphs
Discrete Applied Mathematics
NP completeness of the edge precoloring extension problem on bipartite graphs
Journal of Graph Theory
NP-completeness of list coloring and precoloring extension on the edges of planar graphs
Journal of Graph Theory
Journal of Computer and System Sciences
Testing planarity of partially embedded graphs
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
The LBFS Structure and Recognition of Interval Graphs
SIAM Journal on Discrete Mathematics
On extending a partial straight-line drawing
GD'05 Proceedings of the 13th international conference on Graph Drawing
Extending partial representations of function graphs and permutation graphs
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
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We initiate the study of the computational complexity of the question of extending partial representations of geometric intersection graphs. In this paper we consider classes of interval graphs - given a collection of real intervals that forms an intersection representation of an induced subgraph of an input graph, is it possible to add intervals to achieve an intersection representation of the entire graph? We present an O(n2) time algorithm that solves this problem and constructs a representation if one exists. Our algorithm can also be used to list all nonisomorphic extensions with O(n2) delay. Although the classes of proper and unit interval graphs coincide, the partial representation extension problems differ on them. We present an O(mn) time decision algorithm for partial representation extension of proper interval graphs, but for unit interval graphs the complexity remains open. Finally we show how our methods can be used for solving the problem of simultaneous interval representations. We prove that this problem is fixed-paramater tractable with the size of the common intersection of the input graphs being the parameter.