The ultimate interval graph recognition algorithm?
Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms
A wide-range efficient algorithm for minimal triangulation
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
Journal of Algorithms
On claw-free asteroidal triple-free graphs
Discrete Applied Mathematics
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
LexBFS-Ordering in Asteroidal Triple-Free Graphs
ISAAC '99 Proceedings of the 10th International Symposium on Algorithms and Computation
On Claw-Free Asteroidal Triple-Free Graphs
WG '99 Proceedings of the 25th International Workshop on Graph-Theoretic Concepts in Computer Science
Linear Time Algorithms for Hamiltonian Problems on (Claw, Net)-Free Graphs
WG '99 Proceedings of the 25th International Workshop on Graph-Theoretic Concepts in Computer Science
On the Domination Search Number
WG '00 Proceedings of the 26th International Workshop on Graph-Theoretic Concepts in Computer Science
On Subfamilies of AT-Free Graphs
WG '01 Proceedings of the 27th International Workshop on Graph-Theoretic Concepts in Computer Science
Approximate Distance Labeling Schemes
ESA '01 Proceedings of the 9th Annual European Symposium on Algorithms
On the domination search number
Discrete Applied Mathematics
On linear and circular structure of (claw, net)-free graphs
Discrete Applied Mathematics
Induced matchings in asteroidal triple-free graphs
Discrete Applied Mathematics - Special issue on stability in graphs and related topics
Hereditary dominating pair graphs
Discrete Applied Mathematics
AT-free graphs: linear bounds for the oriented diameter
Discrete Applied Mathematics - Brazilian symposium on graphs, algorithms and combinatorics
Tree-decompositions of small pathwidth
Discrete Applied Mathematics - Structural decompositions, width parameters, and graph labelings (DAS 5)
Discrete Applied Mathematics
Interval completion with few edges
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Feedback vertex set on AT-free graphs
Discrete Applied Mathematics
Efficient algorithms for Roman domination on some classes of graphs
Discrete Applied Mathematics
Discrete Applied Mathematics
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
Tree-decompositions of small pathwidth
Discrete Applied Mathematics - Structural decompositions, width parameters, and graph labelings (DAS 5)
Discrete Applied Mathematics
Pursuing a fast robber on a graph
Theoretical Computer Science
The LBFS Structure and Recognition of Interval Graphs
SIAM Journal on Discrete Mathematics
Separator orders in interval, cocomparability, and AT-free graphs
Discrete Applied Mathematics
Theoretical Computer Science
Localized and compact data-structure for comparability graphs
ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation
Domination search on graphs with low dominating-target-number
WG'05 Proceedings of the 31st international conference on Graph-Theoretic Concepts in Computer Science
Roman domination over some graph classes
WG'05 Proceedings of the 31st international conference on Graph-Theoretic Concepts in Computer Science
Collective tree 1-spanners for interval graphs
WG'05 Proceedings of the 31st international conference on Graph-Theoretic Concepts in Computer Science
ACM Transactions on Algorithms (TALG)
Treelike comparability graphs: characterization, recognition, and applications
WG'04 Proceedings of the 30th international conference on Graph-Theoretic Concepts in Computer Science
Collective tree spanners and routing in AT-free related graphs
WG'04 Proceedings of the 30th international conference on Graph-Theoretic Concepts in Computer Science
Algorithms on graphs with small dominating targets
ISAAC'06 Proceedings of the 17th international conference on Algorithms and Computation
Approximability of the path-distance-width for AT-free graphs
WG'11 Proceedings of the 37th international conference on Graph-Theoretic Concepts in Computer Science
Rainbow connection number and connected dominating sets
Journal of Graph Theory
Induced disjoint paths in AT-Free graphs
SWAT'12 Proceedings of the 13th Scandinavian conference on Algorithm Theory
On the L(h, k)-labeling of co-comparability graphs
ESCAPE'07 Proceedings of the First international conference on Combinatorics, Algorithms, Probabilistic and Experimental Methodologies
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
On the stable degree of graphs
WG'12 Proceedings of the 38th international conference on Graph-Theoretic Concepts in Computer Science
Approximating the path-distance-width for AT-free graphs and graphs in related classes
Discrete Applied Mathematics
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An independent set of three vertices such that each pair is joined by a path that avoids the neighborhood of the third is called an asteroidal triple. A graph is asteroidal triple-free (AT-free) if it contains no asteroidal triples. The motivation for this investigation was provided, in part, by the fact that the AT-free graphs provide a common generalization of interval, permutation, trapezoid, and cocomparability graphs. The main contribution of this work is to investigate and reveal fundamental structural properties of AT-free graphs. Specifically, we show that every connected AT-free graph contains a dominating pair, that is, a pair of vertices such that every path joining them is a dominating set in the graph. We then provide characterizations of AT-free graphs in terms of dominating pairs and minimal triangulations. Subsequently, we state and prove a decomposition theorem for AT-free graphs. An assortment of other properties of AT-free graphs is also provided. These properties generalize known structural properties of interval, permutation, trapezoid, and cocomparability graphs.