Stability number and chromatic number of tolerance graphs
Discrete Applied Mathematics
On the 2-chain subgraph cover and related problems
Journal of Algorithms
Proper and unit tolerance graphs
Discrete Applied Mathematics - Special volume: Aridam VI and VII, Rutcor, New Brunswick, NJ, USA (1991 and 1992)
Trapezoid graphs and generalizations, geometry and algorithms
Discrete Applied Mathematics
Discrete Mathematics
Coloring Algorithms for Tolerance Graphs: Reasoning and Scheduling with Interval Constraints
AISC '02/Calculemus '02 Proceedings of the Joint International Conferences on Artificial Intelligence, Automated Reasoning, and Symbolic Computation
Algorithmic Graph Theory and Perfect Graphs (Annals of Discrete Mathematics, Vol 57)
Algorithmic Graph Theory and Perfect Graphs (Annals of Discrete Mathematics, Vol 57)
Dominating the complements of bounded tolerance graphs and the complements of trapezoid graphs
Discrete Applied Mathematics
A note on tolerance graph recognition
Discrete Applied Mathematics
Max-tolerance graphs as intersection graphs: cliques, cycles, and recognition
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Journal of Graph Theory
A characterization of triangle-free tolerance graphs
Discrete Applied Mathematics
Recognizing bipartite tolerance graphs in linear time
WG'07 Proceedings of the 33rd international conference on Graph-theoretic concepts in computer science
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Tolerance graphs model interval relations in such a way that intervals can tolerate a certain degree of overlap without being in conflict. This class of graphs, which generalizes in a natural way both interval and permutation graphs, has attracted many research efforts since their introduction in [M. C. Golumbic and C. L. Monma, Congr. Numer., 35 (1982), pp. 321-331], as it finds many important applications in constraint-based temporal reasoning, resource allocation, and scheduling problems, among others. In this article we propose the first non-trivial intersection model for general tolerance graphs, given by three-dimensional parallelepipeds, which extends the widely known intersection model of parallelograms in the plane that characterizes the class of bounded tolerance graphs. Apart from being important on its own, this new representation also enables us to improve the time complexity of three problems on tolerance graphs. Namely, we present optimal $\mathcal{O}(n\log n)$ algorithms for computing a minimum coloring and a maximum clique and an $\mathcal{O}(n^{2})$ algorithm for computing a maximum weight independent set in a tolerance graph with $n$ vertices, thus improving the best known running times $\mathcal{O}(n^{2})$ and $\mathcal{O}(n^{3})$ for these problems, respectively.