Abstract
The modular decomposition of a graph G = (V, E) does not contain prime modules if and only if G is a cograph, that is, if no quadruple of vertices induces a simple connected path P4. The cograph editing problem consists in inserting into and deleting from G a set F of edges so that H = (V, E △ F) is a cograph and |F| is minimum. This NP-hard combinatorial optimization problem has recently found applications, e.g., in the context of phylogenetics. Efficient heuristics are hence of practical importance. The simple characterization of cographs in terms of their modular decomposition suggests that instead of editing G one could operate directly on the modular decomposition. We show here that editing the induced P4s is equivalent to resolving prime modules by means of a suitable defined merge operation on the submodules. Moreover, we characterize so-called module-preserving edit sets and demonstrate that optimal pairwise sequences of module-preserving edit sets exist for every non-cograph. This eventually leads to an exact algorithm for the cograph editing problem as well as fixed-parameter tractable (FPT) results when cograph editing is parameterized by the so-called modular-width. In addition, we provide two heuristics with time complexity O(|V|3), resp., O(|V|2).
Highlights
Cographs are of particular interest in computer science because many combinatorial optimization problems that are NP-complete for arbitrary graphs become polynomial-time solvable on cographs [4, 8, 20]
In contrast to the editing approach of [11], we pursue an approach that is modul-preserving in the sense that each module of G is a module of the edited graph G∗. We argue that this property is desirable in the context of orthology detection, because the corrected modular decomposition tree, i.e., the cotree of G∗ has a direct interpretation as event-labeled gene tree [30, 35]
The concept of modular decompositions (MD) is defined for arbitrary graphs G and allows us to present the structure of G in the form of a tree that generalizes the idea of cotrees
Summary
Cographs are of particular interest in computer science because many combinatorial optimization problems that are NP-complete for arbitrary graphs become polynomial-time solvable on cographs [4, 8, 20]. An alternative way of looking at the connection between cographs and their modular decomposition trees is to interpret the destruction of all P4s in a cograph editing algorithm as the resolution of all prime modules in the edited graph G∗. It is sufficient to adjust the neighbors of certain submodules Mi of M to merge the Mi in a way that resolves the prime module M to obtain G∗ In this setting, it seems natural to edit the modular decomposition tree of a graph directly with the aim of converting it step-by-step into the closest modular decomposition tree of a cograph. We show that any graph has an optimal edit set that can be entirely expressed by merging modules that are children of prime modules in the modular decomposition tree. We finish this paper with a short discussion on how the latter method can be used to obtain a simple O(|V |2)-time heuristic
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