A graph is claw-free if no vertex has three pairwise nonadjacent neighbours. In earlier papers of this series we proved that every claw-free graph either belongs to one of several basic classes that we described explicitly, or admits one of a few kinds of decomposition. In this paper we convert this “decomposition” theorem into a theorem describing the global structure of claw-free graphs.

We characterize triangle-free graphs for which there exists a subset of edges that intersects every chordless cycle in an odd number of edges (TF odd-signable graphs). These graphs arise as building blocks of a decomposition theorem (for cap-free odd-signable graphs) obtained by the same authors. We give a polytime algorithm to test membership in this class. This algorithm is itself based on a ...

A bipartite graph is said to be symmetric if it has symmetry of reflecting two vertex sets. This paper investigates matching structure of symmetric bipartite graphs. We first apply the Dulmage-Mendelsohn decomposition to a symmetric bipartite graph. The resulting components, which are matching-covered, turn out to have symmetry. We then decompose a matching-covered bipartite graph via an ear de...

Roughly speaking, the semialgebraic cell decomposition theorem for p-adic numbers describes piecewise the p-adic valuation of p-adic polyno-mials (and more generally of semialgebraic p-adic functions), the pieces being geometrically simple sets, called cells. In this paper we prove a similar cell decomposition theorem to describe piecewise the valuation of analytic functions (and more generally...

Kato's decomposition [14, Theorem 4] is generalized to semi-B-Fredholm operators.

Mostow’s Decomposition Theorem is a refinement of the polar decomposition. It states the following. Let G be a compact connected semi-simple Lie group with Lie algebra g. Given a subspace h of g such that [X, [X,Y ]] ∈ h for all X , Y in h, the complexified group G is homeomorphic to the product G · exp im · exp ih, where m is the orthogonal of h in g with respect to the Killing form. This Theo...

E. Dyer [2] proved that there is no continuous decomposition of a compact irreducible continuum into decomposable continua which is an arc with respect to its elements. The author extends Dyer's result to the plane. Consider a continuous decomposition of the plane into nonseparating compact continua. R. L. Moore [6] has shown that the decomposition space is homeomorphic to the plane. Using Moor...

In this article, we present a measurable version of the spectral decomposition theorem for Z2-action on compact metric space. process, obtain some relationships with shadowing property and k-type weak extending property. Then, introduce definition measure expanding by using properties Borel measure. We also prove one that occurs whenever is invariantly expanding. All supporting results are nece...

A probabilistic Büchi automaton PBA is defined. The probabilistic language (L, p) as defined by the PBA is defined. A decomposition theorem similar to the classical Krohn-Rhodes theorem, but for PBA is proved.

The celebrated Dilworth theorem (Ann. of Math. 5 1 (1950), 161-166) on the decomposition of finite posets was extended by Greene and Kleitman (J. Combin. Theory Ser. A 20 (1976), 41-68). Using the Gallai-Milgram theorem (Acta Sci. Math. 2 1 (1960), 18 l-186) we prove a theorem on acyclic digraphs which contains the Greene-Kleitman theorem. The method of proof is derived from M. Saks’ elegant pr...