Tree decompositions were developed by Robertson and Seymour [21]. Since then algorithms have been developed to solve intractable problems efficiently for graphs of bounded treewidth. In this paper we extend tree decompositions to allow cycles to exist in the decomposition graph; we call these new decompositions plane decompositions because we require that the decomposition graph be planar. Firs...

First I want to say a few words about my graph terminology. If I want to allow loops, I use the adjective reflexive. If I want to allow multiple edges, I use multigraph. Thus, a graph has no loops and no multiple edges. I use valency rather than degree. If we say a graph is 4-valent (or tetravalent), it means it is regular of valency 4, for example. A decomposition of a graph X is a partition o...

We introduce a new decomposition of a graphs into quasi-4-connected components, where we call a graph quasi-4-connected if it is 3-connected and it only has separations of order 3 that separate a single vertex from the rest of the graph. Moreover, we give a cubic time algorithm computing the decomposition of a given graph. Our decomposition into quasi-4-connected components refines the well-kno...

In this paper we give a decomposition of a 4-connected graph G into nonseparating chains, which is similar to an ear decomposition of a 2-connected graph. We also give an O(|V (G)||E(G)|) algorithm that constructs such a decomposition. In applications, the asymptotic performance can often be improved to O(|V (G)|). This decomposition will be used to find four independent spanning trees in a 4co...

A P3-decomposition of a graph is a partition of the edges of the graph into paths of length two. We give a necessary and sufficient condition for a semi-complete multigraph, that is a multigraph with at least one edge between each pair of vertices, to have a P3-decomposition. We show that this condition can be tested in strongly polynomial-time, and that the same condition applies to a larger c...

In this paper we recall the notion of weakly decomposition, we recall some necessary and sufficient conditions for a graph to admit such a decomposition, we introduce the recognition algorithm for the diamond-free graphs which keeps the combinatorial structure of the graph by means of the decomposition, as well as an easy possibility to determine the clique number for the diamond-free graphs.

A factorizing permutation of a given graph is simply a permutation of its vertices of which all decomposition sets are factors [3]. Such a concept appears to play a central role in recent papers dealing with graph decomposition. It is applied here for modular decomposition of directed graphs, and we propose a simple linear algorithm that computes the whole decomposition tree when a factorizing ...

Branch decomposition-based algorithms have been used in practical settings to solve some NP-hard problems like the travelling salesman problem (TSP) and general minor containment. The notions of branch decompositions and branchwidth were introduced by Robertson and Seymour to assist in proving the Graph Minors Theorem. Given a connected graph G and a branch decomposition of G of width k where k...

In this paper we introduce a new form of decomposition of graphs, the (P, Q)-decomposition. We first give an optimal algorithm for finding the 1-decomposition of a graph which is a special case of the (P, Q)-decomposition which was first introduced in [21]. We then examine the connections between the 1-decomposition and well known forms of decomposition of graphs, namely, modular and homogeneou...

A graph is called perfect matching compact (briefly, PM -compact), if its perfect matching graph is complete. Matching-covered PM -compact bipartite graphs have been characterized. In this paper, we show that any PM -compact bipartite graph G with δ(G) ≥ 2 has an ear decomposition such that each graph in the decomposition sequence is also PM -compact, which implies that G is matching-covered.