A Canonical Tree Decomposition for Order Types, and Some Applications

Tangles of graphs have been introduced by Robertson and Seymour in the context of their graph minor theory. Tangles may be viewed as describing “$k$-connected components” of a graph (though in a twisted way). An interesting aspect of tangles is that they can be defined not only for graphs, but more generally for arbitrary connectivity functions (that is, integer-valued submodular and symmetric set functions). However, tangles are difficult to deal with algorithmically. To start with, it is unclear how to represent them, because they are families of separations and, as such, may be exponentially large. Our first contribution is a data structure for representing and accessing all tangles of a graph up to some fixed order. Using this data structure, we can prove an algorithmic version of a very general structure theorem due to Carmesin et al. (for graphs) and Hundertmark (for arbitrary connectivity functions) that yields a canonical tree decomposition whose parts correspond to the maximal tangles. This may be viewed as a generalization of the decomposition of a graph into its 3-connected components.

It is proved that for every number k there exists a number f(k) such that every finite k-connected graph of average degree exceeding f(k) contains an edge whose contraction yields again a k-connected graph. For the proof, tree orders on certain sets of smallest separating sets of the graph in question are constructed. This leads to new canonical tree decompositions as well. © 2005 Wiley Periodicals, Inc. J Graph Theory

Tangles of graphs have been introduced by Robertson and Seymour in the context of their graph minor theory. Tangles may be viewed as describing k-connected components of a graph (though in a twisted way). They play an important role in graph minor theory. An interesting aspect of tangles is that they cannot only be defined for graphs, but more generally for arbitrary connectivity functions (that is, integer-valued submodular and symmetric set functions).However, tangles are difficult to deal with algorithmically. To start with, it is unclear how to represent them, because they are families of separations and as such may be exponentially large. Our first contribution is a data structure for representing and accessing all tangles of a graph up to some fixed order.Using this data structure, we can prove an algorithmic version of a very general structure theorem due to Carmesin, Diestel, Harman and Hundertmark (for graphs) and Hundertmark (for arbitrary connectivity functions) that yields a canonical tree decomposition whose parts correspond to the maximal tangles. (This may be viewed as a generalisation of the decomposition of a graph into its 3-connected components.)

We present here a general framework to design algorithms that compute H-join. For a given bipartite graph H, we say that a graph G admits a H-join decomposition or simply a H-join, if the vertices of G can be partitioned in |H| parts connected as in H. This graph H is a kind of pattern, that we want to discover in G. This framework allows us to present fastest known algorithms for the computation of P 4-join (aka N-join), P 5-join (aka W-join), C 6-join (aka 6-join). We also generalize this method to find a homogeneous pair (also known as 2-module), a pair {M 1,M 2} such that for every vertex x∉(M 1∪M 2) and i∈{1,2}, x is either adjacent to all vertices in M i or to none of them. First used in the context of perfect graphs (Chvátal and Sbihi in Graphs Comb. 3:127–139, 1987), it is a generalization of splits (a.k.a. 1-joins) and of modules. The algorithmics to compute them appears quite involved. In this paper, we describe an O(mn 2)-time algorithm computing all maximal homogeneous pairs of a graph, which not only improves a previous bound of O(mn 3) for finding only one pair (Everett et al. in Discrete Appl. Math. 72:209–218, 1997), but also uses a nice structural property of homogenous pairs, allowing to compute a canonical decomposition tree for sesquiprime graphs (i.e., graphs G having no module and such that for every vertex v∈G, G−v also has no module).