In this paper we present an algorithm for determining the number of spanning trees of a graph G which takes advantage of the structure of the modular decomposition tree of G. Specifically, our algorithm works by contracting the modular decomposition tree of the input graph G in a bottom-up fashion until it becomes a single node; then, the number of spanning trees of G is computed as the product...

Let G be a finite simple graph. Let S ⊆ V (G), its closed interval I[S] is the set of all vertices lying on a shortest path between any pair of vertices of S. The set S is convex if I[S] = S. In this work we define the concept of convex partition of graphs. If there exists a partition of V (G) into p convex sets we say that G is p-convex. We prove that is NP -complete to decide whether a graph ...

Weakly quasi-threshold graphs form a proper subclass of the well-known class of cographs by restricting the join operation. In this paper we characterize weakly quasi-threshold graphs by a finite set of forbidden subgraphs: the class of weakly quasi-threshold graphs coincides with the class of {P4, co-(2P3)}-free graphs. Moreover we give the first linear-time algorithm to decide whether a given...

Phylogenomics heavily relies on well-curated sequence data sets that comprise, for each gene, exclusively 1:1 orthologos. Paralogs are treated as a dangerous nuisance that has to be detected and removed. We show here that this severe restriction of the data sets is not necessary. Building upon recent advances in mathematical phylogenetics, we demonstrate that gene duplications convey meaningful...

In this paper we present an algorithm for determining the number of spanning trees of a graph G which takes advantage of the structure of the modular decomposition tree of G. Specifically, our algorithm works by contracting the modular decomposition tree of the input graph G in a bottom-up fashion until it becomes a single node; then, the number of spanning trees of G is computed as the product...

Suppose V is a finite set and C a collection of subsets of V that contains ∅ and V and is closed under taking intersections. Then C is called a convexity and the ordered pair (V, C) is called an aligned space and the elements of C are referred to as convex sets. For a set S ⊆ V , the convex hull of S relative to C, denoted by CHC(S), is the smallest convex set containing S. A set S of vertices ...

In this paper we consider a graph parameter called contiguity which aims at encoding a graph by a linear ordering of its vertices. We prove that the contiguity of cographs is unbounded but is always dominated by O(logn), where n is the number of vertices of the graph. And we prove that this bound is tight in the sense that there exists a family of cographs on n vertices whose contiguity is Ω(lo...

This chapter deals with two graph classes the Gallai graphs and the anti-Gallai graphs. \Ve construct infinitely many pairs of graphs G and H such that f( G) = r( H). The existence of a finite family offorbidden subgraphs for the Gallai graphs and the ant.i-Gallai graphs to be H-frce, for any fillit.e graph H is proved and the forbidden subgraph characterizations of G for which the Gallai graph...

Let G and H be two cographs. We show that the problem to determine whether H is a retract of G is NP-complete. We show that this problem is fixed-parameter tractable when parameterized by the size of H. When restricted to the class of threshold graphs or to the class of trivially perfect graphs, the problem becomes tractable in polynomial time. The problem is also soluble in linear time when on...

A class of graphs is Kruskalian if Kruskal’s theorem on a well-quasi-ordering of finite trees provides a finite characterization in terms of forbidden induced subgraphs. Let k be a natural number. A graph is a k-cograph if its vertices can be colored with colors from the set {1, . . . ,k} such that for every nontrivial subset of vertices W there exists a partition {W1,W2} into nonempty subsets ...