A Linear-Time Algorithm for Finding Tree-Decompositions of Small Treewidth
SIAM Journal on Computing
The Structure and Number of Obstructions to Treewidth
SIAM Journal on Discrete Mathematics
Safe Reduction Rules for Weighted Treewidth
WG '02 Revised Papers from the 28th International Workshop on Graph-Theoretic Concepts in Computer Science
A New Lower Bound for Tree-Width Using Maximum Cardinality Search
SIAM Journal on Discrete Mathematics
A complete anytime algorithm for treewidth
UAI '04 Proceedings of the 20th conference on Uncertainty in artificial intelligence
New lower and upper bounds for graph treewidth
WEA'03 Proceedings of the 2nd international conference on Experimental and efficient algorithms
Pre-processing for triangulation of probabilistic networks
UAI'01 Proceedings of the Seventeenth conference on Uncertainty in artificial intelligence
Achievable sets, brambles, and sparse treewidth obstructions
Discrete Applied Mathematics
Treewidth: characterizations, applications, and computations
WG'06 Proceedings of the 32nd international conference on Graph-Theoretic Concepts in Computer Science
Treewidth lower bounds with brambles
ESA'05 Proceedings of the 13th annual European conference on Algorithms
SOFSEM'05 Proceedings of the 31st international conference on Theory and Practice of Computer Science
Ultimate generalizations of LexBFS and LEX m
WG'05 Proceedings of the 31st international conference on Graph-Theoretic Concepts in Computer Science
Degree-Based treewidth lower bounds
WEA'05 Proceedings of the 4th international conference on Experimental and Efficient Algorithms
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The Maximum Cardinality Search algorithm visits the vertices of a graph in an order such that at any point, a vertex is visited that has the largest number of visited neighbours. An MCS-ordering of a graph is an ordering of the vertices that can be generated by the Maximum Cardinality Search algorithm. The visited degree of a vertex v in an MCS-ordering is the number of neighbours of v that are before v in the ordering. The MCSLB of an MCS-ordering ψ of G is the maximum visited degree over all vertices v in ψ. Lucena [10] showed that the treewidth of a graph G is at least the MCSLB of any MCS-ordering of G. In this paper, we analyse the maximum MCSLB over all possible MCS-orderings of given graphs G. We give upper and lower bounds for this number for planar graphs. Given a graph G, it is NP-complete to determine if G has an MCS-ordering with MCSLB at least k, for any fixed k≥ 7. Also, this problem does not have a polynomial time approximation algorithm with constant ratio, unless P=NP. Variants of the problem are also shown to be NP-complete. We also propose and experimentally analysed some heuristics for the problem. Several tiebreakers for the MCS algorithm are proposed and evaluated. We also give heuristics that give upper bounds on the maximum MCSLB that an MCS-ordering can obtain which appear to give results close to optimal on several graphs from real life applications.