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A Survey on the Computational Complexity of Coloring Graphs with Forbidden Subgraphs
TLDR
This work surveys known results on the computational complexity of Colouring and $k-Colouring for graph classes that are characterized by one or two forbidden induced subgraphs, and considers a number of variants.
Contraction obstructions for treewidth
Graph Searching and Interval Completion
TLDR
This paper proves monotone properties of searching with the smallest cost, and proves that for any graph G the search cost of G is equal to the smallest number of edges of all interval supergraphs of G.
On the Tractability of Optimization Problems on H-Graphs
TLDR
A more refined complexity analysis of the problems solvable in polynomial time on H-graphs from the perspective of Parameterized Complexity shows that Maximum Independent Set, and Minimum Dominating Set are W[1]-hard being parameterized by the size of $H$ and thesize of the solution.
Backbone colorings for graphs: Tree and path backbones
TLDR
The computational complexity of the problem ‘Given a graph G, a spanning tree T of G, and an integer `, is there a backbone coloring for G and T with at most ` colors?’ jumps from polynomial to NP-complete between ` = 4 (easy for all spanning trees) and ` = 5 (difficult even for spanning paths).
Clique-width: on the price of generality
TLDR
This paper shows that the running time O(nf(k)) of many clique-width based algorithms is essentially the best the authors can hope for (up to a widely believed assumption from parameterized complexity, namely FPT ≠ W[1])---the price they pay for generality.
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