Dynamic distance hereditary graphs using split decomposition
ISAAC'07 Proceedings of the 18th international conference on Algorithms and computation
Lexicographic breadth first search – a survey
WG'04 Proceedings of the 30th international conference on Graph-Theoretic Concepts in Computer Science
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We develop new, easily implemented, optimal recognition algorithms for the cograph, distance hereditary, P4 -reducible and P4-sparse graph families. Cographs are the graphs with no induced path on four vertices (P 4), distance hereditary graphs are named after their characterizing property that the distance between any two connected vertices x and y of an induced subgraph equals their distance in the original graph, P4-reducible graphs are the graphs such that no vertex belongs to more than one P4, and P 4-sparse graphs are the graphs such that no set of five vertices induces more than one P4. Each graph family has a rooted tree representation, which defines a decomposition scheme: for Cographs, modular decomposition; for distance hereditary graphs, split decomposition; and for P4-sparse and P4-reducible, a simplification of homogeneous decomposition. Consequently, for these families, the recognition algorithms are also decomposition algorithms. Structurally, each algorithm adheres to the following simple paradigm; order the vertices using a multisweep LexBFS approach, check a neighbourhood property and report a certificate verifying inclusion or exclusion in(from) the family. The algorithms are based upon new graph characterizations and a new search, LexBFS−, a variant of Lexicographic Breadth First Search (LexBFS); LexBFS− operates on the complement using only the edges of the given graph. The neighbourhood check, for each graph family, is based on a new characterization of the graph class with respect to LexBFS. The final step returns a certificate in the form of a tree representation if the graph is in the family, and a forbidden induced subgraph otherwise. Each step is linear in the number of edges and vertices of the graph combining for an optimal algorithm. These algorithms are significant because of their simplicity, linear time complexity and potential to be extended to recognize other graph classes such as permutation graphs, HHD-free graphs, P4-extendible etc., and ultimately, to solve the related decomposition problems.