About complexity of complex networks

We derive complexity estimates for two classes of deterministic networks: the Boolean networks S(Bn, m), which compute the Boolean vector-functions Bn, m, and the classes of graphs $G(V_{P_{m,\,l}}, E)$ G ( V P m , l , E ) , with overlapping communities and high density. The latter objects are well suited for the synthesis of resilience networks. For the Boolean vector-functions, we propose a synthesis of networks on a NOT, AND, and OR logical basis and unreliable channels such that the computation of any Boolean vector-function is carried out with polynomial information cost.All vertexes of the graphs $G(V_{P_{m,\,l}}, E)$ G ( V P m , l , E ) are labeled by the trinomial (m2±l,m)-partitions from the set of partitions Pm, l. It turns out that such labeling makes it possible to create networks of optimal algorithmic complexity with highly predictable parameters. Numerical simulations of simple graphs for trinomial (m2±l,m)-partition families (m=3,4,…,9) allow for the exact estimation of all commonly known topological parameters for the graphs. In addition, a new topological parameter—overlapping index—is proposed. The estimation of this index offers an explanation for the maximal density value for the clique graphs $G(V_{P_{m,\,l}}, E)$ G ( V P m , l , E ) .

On probe 2-clique graphs and probe diamond-free graphs

Graph Theory International audience Given a class G of graphs, probe G graphs are defined as follows. A graph G is probe G if there exists a partition of its vertices into a set of probe vertices and a stable set of nonprobe vertices in such a way that non-edges of G, whose endpoints are nonprobe vertices, can be added so that the resulting graph belongs to G. We investigate probe 2-clique graphs and probe diamond-free graphs. For probe 2-clique graphs, we present a polynomial-time recognition algorithm. Probe diamond-free graphs are characterized by minimal forbidden induced subgraphs. As a by-product, it is proved that the class of probe block graphs is the intersection between the classes of chordal graphs and probe diamond-free graphs.