Graph minors, decompositions and algorithms

Branch and Tree Decomposition Techniques for Discrete Optimization

This chapter gives a general overview of two emerging techniques for discrete optimization that have footholds in mathematics, computer science, and operations research: branch decompositions and tree decompositions. Branch decompositions and tree decompositions along with their respective connectivity invariants, branchwidth and treewidth, were first introduced to aid in proving the Graph Mino...

A combinatorial optimization algorithm for solving the branchwidth problem

In this paper, we consider the problem of computing an optimal branch decomposition of a graph. Branch decompositions and branchwidth were introduced by Robertson and Seymour in their series of papers that proved the Graph Minors Theorem. Branch decompositions have proven to be useful in solving many NP-hard problems, such as the traveling salesman, independent set, and ring routing problems, b...

Simplicial Minors and Decompositions of Graphs

The monadic second-order logic of graphs III : tree-decompositions, minors and complexity issues

We relate the tree-decompositions of hypergraphs introduced by Robertson and Seymour to the finite and infinité algebraic expressions introduced by Bauderon and Courcelle. We express minor inclusion in monadic second-order logic, and we obtain grammatical characterizations of certain sets of graphs defined by excluded minors. We show how tree-decompositions can be used to construct quadratic al...

Computing Bounded Path Decompositions in Logspace

The complexity of Graph Isomorphism (GI) is a long-standing open problem. GI is not known to be solvable in polynomial time and is unlikely to be NP-complete [BHZ87, Sch88]. Polynomial-time algorithms are known for GI of several special classes of graphs. One such interesting class is graphs excluding a fixed minor. Any family of graphs that is closed under taking minors falls into this class. ...