Practical and Efficient Split Decomposition via Graph-Labelled Trees

Split decomposition of graphs was introduced by Cunningham (under the name join decomposition) as a generalization of the modular decomposition. This paper undertakes an investigation into the algorithmic properties of split decomposition. We do so in the context of graph-labelled trees (GLTs), a new combinatorial object designed to simplify its consideration. GLTs are used to derive an incremental characterization of split decomposition, with a simple combinatorial description, and to explore its properties with respect to Lexicographic Breadth-First Search (LBFS). Applying the incremental characterization to an LBFS ordering results in a split decomposition algorithm that runs in time $O(n+m)\alpha(n+m)$, where $\alpha$ is the inverse Ackermann function, whose value is smaller than 4 for any practical graph. Compared to Dahlhaus' linear-time split decomposition algorithm [Dahlhaus'00], which does not rely on an incremental construction, our algorithm is just as fast in all but the asymptotic sense and full implementation details are given in this paper. Also, our algorithm extends to circle graph recognition, whereas no such extension is known for Dahlhaus' algorithm. The companion paper [Gioan et al.] uses our algorithm to derive the first sub-quadratic circle graph recognition algorithm

Forbidden minors characterization of partial 3-trees

AbstractA k-tree is formed from a k-complete graph by recursively adding a vertex adjacent to all vertices in an existing k-complete subgraph. The many applications of partial k-trees (subgraphs of k-trees) have motivated their study from both the algorithmic and theoretical points of view. In this paper we characterize the class of partial 3-trees by its set of four minimal forbidden minors (H is a minor of G if H can be obtained from G by a finite sequence of edge-extraction and edge-contradiction operations.

Ki-covers I: Complexity and polytopes

AbstractA Ki in a graph is a complete subgraph of size i. A Ki-cover of a graph G(V, E is a set C of Ki − 1's of G such that every Ki in G contains at least one Ki − 1 in C. Thus a K2-cover is a vertex cover. The problem of determining whether a graph has a Ki-cover (i ⩾ 2) of cardinality ⩽k is shown to be NP-complete for graphs in general. For chordal graphs with fixed maximum clique size, the problem is polynomial; however, it is NP-complete for arbitrary chordal graphs when i ⩾ 3. The NP-completeness results motivate the examination of some facets of the corresponding polytope. In particular we show that various induced subgraphs of G define facets of the Ki-cover polytope. Further results of this type are also produced for the K3-cover polytope. We conclude by describing polynomial algorithms for solving the separation problem for some classes of facets of the Ki-cover polytope

Vertex ordering characterizations of graphs of bounded asteroidal number

Asteroidal Triple-free (AT-free) graphs have received considerable attention due to their inclusion of various important graphs families, such as interval and cocomparability graphs. The asteroidal number of a graph is the size of a largest subset of vertices such that the removal of the closed neighborhood of any vertex in the set leaves the remaining vertices of the set in the same connected component. (AT-free graphs have asteroidal number at most 2.) In this article, we characterize graphs of bounded asteroidal number by means of a vertex elimination ordering, thereby solving a long-standing open question in algorithmic graph theory. Similar characterizations are known for chordal, interval, and cocomparability graphs

Unified View of Graph Searching and LDFS-Based Certifying Algorithms

International audienc