(Special issue FPSAC 2002)International audienceThe main results of the paper are two dual algorithms bijectively mapping the set of spanning trees with internal activity 1 and external activity 0 of an ordered graph onto the set of acyclic orientations with adjacent unique source and sink. More generally, these algorithms extend to an activity-preserving correspondence between spanning trees and orientations. For certain linear orderings of the edges, they also provide a bijection between spanning trees with external activity 0 and acyclic orientations with a given unique sink. This construction uses notably an active decomposition for orientations of a graph which extends the notion of components for acyclic orientations with unique given sink

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In this paper, we revisit the split decomposition of graphs and give new combinatorial and algorithmic results for the class of totally decomposable graphs, also known as the distance hereditary graphs, and for two non-trivial subclasses, namely the cographs and the 3-leaf power graphs. Precisely, we give strutural and incremental characterizations, leading to optimal fullydynamic recognition algorithms for vertex and edge modifications, for each of these classes. These results rely on the new combinatorial framework of graph-labelled trees used to represent the split decomposition of general graphs. The point of the paper is to use bijections between the aforementioned graph classes and graph-labelled trees whose nodes are labelled by cliques and stars. We mention that this bijective viewpoint yields directly an intersection model for the class of distance hereditary graphs. IntroductionThe 1-join composition and its complementary operation, the split decomposition, range among the classical operations in graph theory. It was introduced by Cunningham and Edmonds [8,9] in the early 80's and has, since then, been used in various contexts such as perfect graph theory [30], circle graphs [5], clique-with [13] or rank-width [38]. The first polynomial time algorithm to compute the split decomposition of a graph, proposed in [8], runs O(n 3 ) time complexity. It was later improved by Ma and Spinrad [35] who described an O(n 2 ) time algorithm. So far Dahlhaus' linear time algorithm [17] is the fastest. Also, we mention the recent work [11] which nicely reformulates underlying routines from [17].Roughly speaking, a split is a bipartition of the vertices of a graph satisfying certain properties (see Definition 2.7). Computing the split decomposition of a graph consists in recursively decompose that graph according to bipartitions that are splits. This process naturally yields a (split) decomposition tree [8,9] which represents the used bipartitions. However such a tree does not keep *