There are many graph problems that can be solved in linear or polynomial time with a dynamic programming algorithm when the input graph has bounded treewidth. For combinatorial optimization problems, this is a useful approach for obtaining fixed-parameter tractable algorithms. Starting from trees and series-parallel graphs, we introduce the concepts of treewidth and tree decompositions, and illustrate the technique with the Weighted Independent Set problem as an example. The paper surveys some of the latest developments, putting an emphasis on applicability, on algorithms that exploit tree decompositions, and on algorithms that determine or approximate treewidth and find tree decompositions with optimal or close to optimal treewidth. Directions for further research and suggestions for further reading are also given.

Combinatorial Optimization on Graphs of Bounded Treewidth

The complexity of scheduling independent two-processor tasks on dedicated processors

The problems of finding values of the chromatic number and the chromatic index of a graph are NP-hard even for some restricted classes of graphs. Every outerplanar graph has an associated tree structure which facilitates algorithmic treatment. Using that structure, we give an efficient algorithm to color the vertices of an outerplanar graph with the minimum number of colors. We also establish algorithmically the value of the chromatic index of an outerplanar graph. Our algorithms are based on systematic coloring of elements (vertices and edges, respectively) of adjacent faces.

Efficient vertex- and edge-coloring of outerplanar graphs

A linear-time algorithm for finding a minimum spanning pseudoforest

Optimally edge-colouring outerplanar graphs is in NC

Linear algorithms for edge-coloring trees and unicyclic graphs

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Linear algorithms for edge-coloring trees and unicyclic graphs