Finding occurrences of protein complexes in protein–protein interaction graphs

In the context of comparative analysis of protein–protein interaction graphs, we use a graph-based formalism to detect the preservation of a given protein complex G in the protein–protein interaction graph H of another species with respect to (w.r.t.) orthologous proteins. Two problems are considered: the Exact-(μG,μH)(μG,μH)-Matching problem and the Max-(μG,μH)(μG,μH)-Matching problems, where μGμG (resp. μHμH) denotes in both problems the maximum number of orthologous proteins in H (resp. G) of a protein in G (resp. H). Following [I. Fagnot, G. Lelandais, S. Vialette, Bounded list injective homomorphism for comparative analysis of protein–protein interaction graphs, Journal of Discrete Algorithms 6 (2) (2008) 178–191], the Exact-(μG,μH)(μG,μH)-Matching problem asks for an injective homomorphism of G to H w.r.t. orthologous proteins. The optimization version is called the Max-(μG,μH)(μG,μH)-Matching problem and is concerned with finding an injective mapping of a graph G to a graph H w.r.t. orthologous proteins that matches as many edges of G   as possible. For both problems, we essentially focus on bounded degree graphs and extremal small values of parameters μGμG and μHμH.