The notion of treewidth has seen to be a powerful vehicle for many graph algorithmic studies. This survey paper wants to give an overview of many classes of graphs that can be seen to have a uniform upper bound on the treewidth of graphs in the class. Also, some mutual relations between such classes are discussed.

this paper are (i) a construction by which, for a given bounded treewidth, a general MS graph property P is transformed to an MS binary tree property r(P), and a general labeled graph G with a suitable tree-decomposition is transformed to a labeled binary tree T(G) in time linear in the number of vertices of G and in such a way that P holds for G if and only if r(P) holds for T(G). This allows us, using techniques developed by Doner [20] and Thatcher and Wright [42], to compile a tree automaton which decides the MS-problem r(P) on the tree T(G) (and thus also P on the graph G) in linear time, and (ii) a procedure whereby such an automaton for a MS formula with free variables is modified to solve a related EMS problem involving counting

This paper gives an overview of several results and techniques for graphs algorithms that compute the treewidth of a graph or that solve otherwise intractable problems when restricted graphs with bounded treewidth more efficiently. Also, several results on graph minors are reviewed.

An algorithm is developed for finding a close to optimal junction tree of a given graph G.

In this paper we discuss the problem of finding the treewidth and pathwidth of permutation graphs. If G[r] is a permutation graph with treewidth k, then we show that the pathwidth of G[r] is at most 2k, and we give an algo- rithm which constructs a path-decomposition with width at most 2k in time O(nk). We assume that the permutation r is given. For permutation graphs of which the treewidth is bounded by some constant, this result, together with previous results ([9], [15]), shows that the treewidth and pathwidth can be computed in linear time.

In this paper we give, for all constants k, l, explicit algorithms, that given a graph G = (V; E) with a tree-decomposition of G with treewidth at most l, decide whether the treewidth (or pathwidth) of G is at most k, and if so, find a tree-decomposition or (path-decomposition) of G of width at most k, and that use O(|V|) time. In contrast with previous solutions, our algorithms do not rely on non-constructive reasoning, and are single exponential in k and l. This result can be combined with a result of Reed [37], yielding explicit O(n log n) algorithms for the problem, given a graph G, to determine whether the treewidth (or pathwidth) of G is at most k, and if so, to find a tree- (or path-)decomposition of width at most k (k constant). Also, Bodlaender [13] has used the result of this paper to obtain linear time algorithms for these problems. We also show that for all constants k, there exists a polynomial time algorithm, that, when given a graph G = (V; E) with treewidth k, computes the pathwidth of G and a minimum path decomposition of G.

We prove that, for any fixed k, one can construct a linear time algorithm that checks if a graph has branchwidth k and, if so, outputs a branch decomposition of minimum width. 1 Introduction This paper considers the problem of finding branch decompositions of graphs with small branchwidth. The notion of branchwidth has a close relationship to the more well-known notion of treewidth, a notion that has come to play a large role in many recent investigations in algorithmic graph theory. (See Section 2 for definitions of treewidth and branchwidth.) One reason for the interest in this notion is that many graph problems can be solved by linear time algorithms, when the inputs are restricted to graphs with some uniform upper bound on their treewidth. Most of these algorithms first try to find a tree decomposition of small width, and then utilize the advantages of the tree structure of the decomposition (see [1], [4]). The branchwidth of a graph differs from its treewidth by at most a multipl...

In this paper we investigate both the structure of graphs with branchwidth at most three, as well as algorithms to recognise such graphs. We show that a graph has branchwidth at most three, if and only if it has treewidth at most three and does not contain the three-dimensional binary cube graph as a minor. A set of four graphs is shown to be the obstruction set of graphs with branchwidth at most three. We give a safe and complete set of reduction rules for the graphs with branchwidth at most three. Using this set, a linear time algorithm is given that checks if a given graph has branchwidth at most three, and, if so, outputs a minimum width branch decomposition. Keywords: graph algorithms, branchwidth, obstruction set, graph minor, reduction rule. 1 Introduction This paper considers the study of the graphs with branchwidth at most three. The notion of branchwidth has a close relationship to the more well-known notion of treewidth, a notion that has come to play a large role in many ...

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A graph G has tree-width at most w if it admits a tree-decomposition of width ≤ w. It is known that once we have a tree-decomposition of a graph G of bounded width, many NP-hard problems can be solved for G in linear time. For w ≤ 3 we give a linear-time algorithm for finding such a decomposition and for a general fixed w we obtain a probabilistic algorithm with execution time O(n log 2 n + n log n | log p|), which for a graph G on n vertices and a real number p> 0 either finds a tree-decomposition of width ≤ 6w or answers that the tree-width of G is ≥ w; this second answer may be wrong but with probability at most p. The second result is based on a separator technique which may be of independent interest.