Inverse problems for the number of maximal independent sets

We study the following inverse graph-theoretic problem: how many vertices should a graph have given that it has a specified value of some parameter. We obtain asymptotic for the minimal number of vertices of the graph with the given number n of maximal independent sets for a class of natural numbers that can be represented as concatenation of periodic binary words. Problems of estimating various graph invariants play the central role in quantitative graph theory. Among the most studied invariants are connectivity, chromatic number, girth, independence number, maximal clique size, number of independent sets etc. As well as forward problems, inverse problems also are of interest. They generally can be stated as follows: find a graph (or prove its existence) that have the desired value of some parameter. The classical problem of this kind is finding a graph with the given degree sequence [2, 3]. For a long time it was not known if there was only a finite number of naturals not being the Wiener index of trees. [5, 6]. An analogous question considering the number of independent sets in trees, asked in [4], is not yet solved, whereas some other parameters of trees are better studied (e.g. [1]). We now state the problems coevered in this paper in their general form. Let G be a family of graphs, and let φ : G → S and ψ : G → T be arbitrary functionals on G. The existential inverse problem for the pair (G, φ) may be stated as follows: “describe all s ∈ S for which there exists a graph G ∈ G having φ(G) = s”. Let S be the set of all values of ψ for all graphs in G. For S ⊆ N we call G to be strongly φ-complete, if for every s ∈ S there is G ∈ G such that φ(G) = s. If such G ∈ G exists for all large enough s ∈ S, then we say that G is weakly φ-complete, or just φ-complete. If φ(G) = s then we say that s is realized by G. If the existential inverse problem is solved positively, we can consider the optimizational inverse problem for the triple (G, φ, ψ): “for a given s ∈ S find LGφ,ψ(s) = inf{ψ(G) | G ∈ G, φ(G) = s}”. As the problem of finding L exactly is too hard, it is natural to consider only the asymptotic behavior of LGφ,ψ(s) for φ-complete families of graphs. If G is a class of all graphs, we shorten the notation LGφ,ψ(s) to Lφ,ψ(s). Denote by ι(G) the number of all independent sets (i. s.) of vertices in G, and by ιm(G) the number of maximal-by-inclusion i.s. (m. i. s.) in G. Finally, by ιM (G) we denote the number of maximum independent sets in G. We write ν(G) and ǫ(G) for the number of vertices and edges in G respectively. The families of bipartite graphs and forests are denoted by B and F respectively. We write Kr and Pr for complete graphs and paths on r vertices. Kr,s denotes complete bipartite graph, r and s being the sizes of its parts. K ′ r,r stands for the corona-graph, which can be constructed by deleting edges of some perfect matching from Kr,r. The sets of