Linear time algorithms for NP-hard problems restricted to partial k-trees
Discrete Applied Mathematics
Handle-rewriting hypergraph grammars
Journal of Computer and System Sciences
The expression of graph properties and graph transformations in monadic second-order logic
Handbook of graph grammars and computing by graph transformation
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Edge dominating set and colorings on graphs with fixed clique-width
Discrete Applied Mathematics
On the complexity of the maximum cut problem
Nordic Journal of Computing
A Mickey-Mouse Decomposition Theorem
Proceedings of the 4th International IPCO Conference on Integer Programming and Combinatorial Optimization
On the Clique-Width of Graphs in Hereditary Classes
ISAAC '02 Proceedings of the 13th International Symposium on Algorithms and Computation
How to Solve NP-hard Graph Problems on Clique-Width Bounded Graphs in Polynomial Time
WG '01 Proceedings of the 27th International Workshop on Graph-Theoretic Concepts in Computer Science
Journal of Combinatorial Theory Series B
Algorithms for vertex-partitioning problems on graphs with fixed clique-width
Theoretical Computer Science
Decomposition of odd-hole-free graphs by double star cutsets and 2-joins
Discrete Applied Mathematics - Brazilian symposium on graphs, algorithms and combinatorics
On the Relationship Between Clique-Width and Treewidth
SIAM Journal on Computing
IEEE/ACM Transactions on Computational Biology and Bioinformatics (TCBB)
Even and odd holes in cap-free graphs
Journal of Graph Theory
Even-hole-free graphs part I: Decomposition theorem
Journal of Graph Theory
Claw-free graphs. IV. Decomposition theorem
Journal of Combinatorial Theory Series B
Even-hole-free graphs that do not contain diamonds: A structure theorem and its consequences
Journal of Combinatorial Theory Series B
A structure theorem for graphs with no cycle with a unique chord and its consequences
Journal of Graph Theory
Graphs of separability at most two: structural characterizations and their consequences
IWOCA'10 Proceedings of the 21st international conference on Combinatorial algorithms
Substitution and Χ-boundedness
Journal of Combinatorial Theory Series B
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We introduce graphs of separability at mostk as graphs in which every two non-adjacent vertices are separated by a set of at most k other vertices. Graphs of separability at most k arise in connection with the Parsimony Haplotyping problem from computational biology. For k@?{0,1}, the only connected graphs of separability at most k are complete graphs and block graphs, respectively. For k=3, graphs of separability at most k form a rich class of graphs containing all graphs of maximum degree k. We prove several characterizations of graphs of separability at most 2, which generalize complete graphs, cycles and trees. The main result is that every connected graph of separability at most 2 can be constructed from complete graphs and cycles by pasting along vertices or edges, and vice versa, every graph constructed this way is of separability at most 2. The structure theorem has nice algorithmic implications-some of which cannot be extended to graphs of higher separability-however certain optimization problems remain intractable on graphs of separability 2. We then characterize graphs of separability at most 2 in terms of minimal forbidden induced subgraphs and minimal forbidden induced minors. Finally, we discuss the possibilities of extending these results to graphs of higher separability.