Symbolic ultrametrics define edge-colored complete graphs Kn and yield a simple tree representation of Kn. We discuss, under which conditions this idea can be generalized to find a symbolic ultrametric that, in addition, distinguishes between edges and non-edges of arbitrary graphs G = (V,E) and thus, yielding a simple tree representation of G. We prove that such a symbolic ultrametric can only...

For every cograph there exist bases of the eigenspaces for the eigenvalues 0 and −1 that consist only of vectors with entries from {0, 1,−1}, a property also exhibited by other graph classes. Moreover, the multiplicities of the eigenvalues 0 and −1 of a cograph can be determined by counting certain vertices of the associated cotree.

The class of cographs, or complement-reducible graphs, arises naturally in many different areas of applied mathematics and computer science. We show that the problem of finding a maximum matching in a cograph can be solved optimally in parallel by reducing it to parenthesis matching. With an $n$-vertex cograph $G$ represented by its parse tree as input, our algorithm finds a maximum matching in...

This paper presents an optimal fully-dynamic recognition algorithm for directed cographs. Given the modular decomposition tree of a directed cograph G, the algorithm supports arc and vertex modification (insertion or deletion) in O(d) time where d is the number of arcs involved in the operation. Moreover, if the modified graph remains a directed cograph, the modular tree decomposition is update...

A cograph is a P4–free graph. We first give a short proof of the fact that 0 (−1) belongs to the spectrum of a connected cograph (with at least two vertices) if and only if it contains duplicate (resp. coduplicate) vertices. As a consequence, we next prove that the polynomial reconstruction of graphs whose vertex–deleted subgraphs have the second largest eigenvalue not exceeding √ 5−1 2 is uniq...

We exemplify an optimisation criteria for divide-and-conquer algorithms with a technique called generic competitive graph search. The technique is then applied to solve two problems arising from biocomputing, so-called Common Connected Components and Cograph Sandwich. The first problem can be defined as follows: given two graphs on the same set of n vertices, find the coarsest partition of the ...

We give an abstract formulation of proof-structures and nets for pomset logic which is shown to be equivalent to the original one with links. Here a proof-structure is described as a directed R&B-cograph, that is an edge bicoloured graph, one colour being a perfect matching and the other being a directed cograph — directed cographs are are a simple generalization of cographs and series-parallel...

In 1985, Corneil, Perl and Stewart CPS85] gave a linear incremental algorithm to recognize cographs (graphs with no induced P4). When this algorithm stops, either the initial graph is a cograph and the cotree of the whole graph has been built, or the initial graph is not a cograph and this algorithm ends up with a vertex v and a cotree cot such that v cannot be inserted in cot; so the input gra...

In this paper, we establish structural properties for the class of complement reducible graphs or cographs, which enable us to describe efficient parallel algorithms for recognizing cographs and for constructing the cotree of a graph if it is a cograph; if the input graph is not a cograph, both algorithms return an induced P4. For a graph on n vertices andm edges, both our cograph recognition a...