Incremental modular decomposition
Journal of the ACM (JACM)
Recognizing P4-sparse graphs in linear time
SIAM Journal on Computing
A tree representation for P4-sparse graphs
Discrete Applied Mathematics
P4-trees and substitution decomposition
Discrete Applied Mathematics
On extendedP4-reducible and extendedP4-sparse graphs
Theoretical Computer Science
Graph classes: a survey
Modular decomposition and transitive orientation
Discrete Mathematics - Special issue on partial ordered sets
Fully dynamic algorithms for chordal graphs
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
Efficient and practical algorithms for sequential modular decomposition
Journal of Algorithms
Introduction to Algorithms
Linear-Time Representation Algorithms for Proper Circular-Arc Graphs and Proper Interval Graphs
SIAM Journal on Computing
A Fully Dynamic Algorithm for Recognizing and Representing Proper Interval Graphs
SIAM Journal on Computing
On-Line Recognition of Interval Graphs in O(m + nlog n) Time
Selected papers from the 8th Franco-Japanese and 4th Franco-Chinese Conference on Combinatorics and Computer Science
A New Linear Algorithm for Modular Decomposition
CAAP '94 Proceedings of the 19th International Colloquium on Trees in Algebra and Programming
A fully dynamic algorithm for modular decomposition and recognition of cographs
Discrete Applied Mathematics - The 1st cologne-twente workshop on graphs and combinatorial optimization (CTW 2001)
Fully dynamic algorithm for recognition and modular decomposition of permutation graphs
WG'05 Proceedings of the 31st international conference on Graph-Theoretic Concepts in Computer Science
Fully-Dynamic recognition algorithm and certificate for directed cographs
WG'04 Proceedings of the 30th international conference on Graph-Theoretic Concepts in Computer Science
An optimal, edges-only fully dynamic algorithm for distance-hereditary graphs
STACS'07 Proceedings of the 24th annual conference on Theoretical aspects of computer science
A fully dynamic algorithm for the recognition of P4-sparse graphs
Theoretical Computer Science
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We consider the dynamic recognition problem for the class of P4-sparse graphs: the objective is to handle edge/vertex additions and deletions, to recognize if each such modification yields a P4-sparse graph, and if yes, to update a representation of the graph. Our approach relies on maintaining the modular decomposition tree of the graph, which we use for solving the recognition problem. We establish conditions for each modification to yield a P4-sparse graph and obtain a fully dynamic recognition algorithm which handles edge modifications in O(1) time and vertex modifications in O(d) time for a vertex of degree d. Thus, our algorithm implies an optimal edges-only dynamic algorithm and a new optimal incremental algorithm for P4-sparse graphs. Moreover, by maintaining the children of each node of the modular decomposition tree in a binomial heap, we can handle vertex deletions in O(log n) time, at the expense of needing O(log n) time for each edge modification and O(d log n) time for the addition of a vertex adjacent to d vertices.