Split decomposition and graph-labelled trees: Characterizations and fully dynamic algorithms for totally decomposable graphs

Abstract

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 *

Solving Problems on Graphs of High Rank-Width

“…3). The contents of this subsection are based on the original work of Cunningham [13] as well as on more recent results by several authors [25][26][27].…”

“…When clear from the context, we may use u as a shorthand for G(u) ∈ F; for instance, we use V (u) to denote V (G(u)) and we say that an edge of T incident to u is incident to the vertex of G(u) mapped to it. Graph-labeled trees were introduced by Gioan and Paul [25,26], and in the following paragraphs we recall some useful definitions and facts that also appeared in follow-up work [27].…”

“…A graph-labeled tree is reduced if all its labels are either prime or degenerate, and no node-join of two cliques or two stars S 1 and S 2 where the center of S 1 is adjacent to a leaf of S 2 is not possible. It is known that for every connected graph G, there exists a unique reduced graph-labeled tree (T, F) such that G = Gr(T, F) [13,[25][26][27]; this unique reduced graph-labeled tree is called the split-tree and is denoted ST(G). …”

Exponential Bounds on Graph Enumerations from Vertex Incremental Characterizations

“…A vertex incremental characterization of a class of graphs A is the necessary and sufficient conditions under which adding a vertex v to a graph from A would produce another graph from A [18].…”

“…Two well-known examples of such decompositions are the modular decomposition, and the split decomposition [12,18]. The latter was recently used by Chauve et al [7,8], to obtain an exact enumeration of an important class of (perfect) graphs, the distance hereditary graphs.…”