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.

💡 Research Summary

The paper tackles an “inverse” question in graph theory: given a prescribed number n of maximal independent sets (MIS), what is the smallest possible order of a graph that realizes exactly n MIS? While forward problems—computing parameters such as independence number, chromatic number, or the number of MIS from a given graph—are classical, the reverse direction has received little attention. The authors focus on the function
 f(n) = min {|V(G)| : i(G)=n},
where i(G) denotes the number of maximal independent sets of G.

Background and simple bounds.
A trivial information‑theoretic lower bound follows from the fact that each MIS is a distinct subset of vertices, so 2^{|V(G)|} ≥ n, yielding f(n) ≥ ⌈log₂ n⌉. Existing constructions for arbitrary n (e.g., using bipartite graphs or trees) give an upper bound of the form f(n) ≤ 2⌈log₂ n⌉ + O(1). This leaves a substantial gap, especially for large n.

Main contribution – periodic binary words.
The authors identify a large and natural subclass of integers that can be written as a concatenation of a fixed binary block w repeated k times:
 n = (w)^{k} = binary(w)·2^{p(k‑1)} + … + binary(w),
where p = |w| is the length of the block. Numbers of this form include powers of two, numbers with repeating patterns such as 101010…, and many others. For such n the paper proves an asymptotically tight estimate:

 f(n) = (1 + o(1))·log₂ n.

In other words, for periodic binary n the minimal vertex count matches the information‑theoretic lower bound up to a vanishing additive term.

Construction – the “block graph” B(w).
Given a binary word w = w₁w₂…w_p, the authors build a gadget B(w) as follows. For each position i they create a layer X_i of vertices. If w_i = 1, the vertices of X_i are made pairwise adjacent, forming a clique; if w_i = 0, X_i remains an independent set. Between consecutive layers all possible edges are added, turning the whole structure into a directed‑like cascade that prevents an independent set from spanning more than one layer. Consequently, any maximal independent set of B(w) must consist of exactly one vertex from each “1‑layer” and none from the “0‑layers”. The number of distinct MIS in B(w) is therefore 2^{p}, matching the number of binary strings of length p.

Repeating the gadget – G(w,k).
To obtain n = 2^{p·k} MIS, the authors concatenate k copies of B(w) in a cyclic fashion, adding the same inter‑layer edges between the last copy and the first. The resulting graph G(w,k) has p·k + O(p) vertices. Because each copy contributes an independent choice of a maximal independent set, the total number of MIS in G(w,k) is (2^{p})^{k} = 2^{p·k} = n. The construction is explicit, polynomial‑time, and yields a graph whose order is exactly (1 + o(1))·log₂ n.

General n via block decomposition.
For numbers that are not perfectly periodic, the authors propose a decomposition algorithm: write n’s binary expansion as a concatenation of the longest possible periodic blocks w₁^{k₁}, w₂^{k₂}, …, then build the corresponding block graphs for each piece and connect them with a single edge between successive pieces. This “divide‑and‑conquer” approach reduces the vertex count compared with the naïve bipartite construction, improving the known upper bound from 2⌈log₂ n⌉ + O(1) to roughly (1.9)⌈log₂ n⌉ + O(1) in the worst case, and achieving the optimal (1 + o(1))·log₂ n for the periodic subclass.

Proof sketch.
The lower bound is immediate: any family of n distinct subsets of a ground set of size m must satisfy 2^{m} ≥ n. For the upper bound, the authors verify two properties of G(w,k): (i) every maximal independent set corresponds uniquely to a choice of one vertex from each “1‑layer” in each copy of B(w); (ii) the inter‑copy edges guarantee maximality, i.e., no set can be extended across copies. By counting the independent choices they obtain i(G(w,k)) = n, and by counting vertices they obtain |V(G(w,k))| = p·k + O(p). Since n = 2^{p·k}, we have |V| = (1 + o(1))·log₂ n.

Experimental validation.
The authors implemented the construction and tested it on 10 000 random integers up to 10⁶. They compared the vertex count of their graphs with those produced by the best known generic constructions (complete bipartite graphs, trees, and the “binary‑tree” gadget). For numbers with a clear periodic pattern, the new method reduced the vertex count by an average of 12 % and never performed worse than the baseline. For arbitrary numbers the average improvement was about 6 %, confirming that the block‑decomposition heuristic is beneficial even when perfect periodicity is absent.

Implications and future work.
The results have two immediate implications. First, they provide a near‑optimal solution to a natural inverse design problem, which can be useful in network synthesis where the number of maximal independent sets corresponds to the number of fault‑tolerant configurations or resource‑allocation scenarios. Second, the methodology—identifying a class of numbers with a regular binary structure, building a tailored gadget, and then extending via concatenation—suggests a general template for other inverse problems, such as fixing the chromatic number, clique number, or domination number. The authors outline a research agenda to explore these directions, expecting that similar “periodic‑word” constructions will yield tight bounds for many other graph parameters.

In summary, the paper establishes that for a broad and natural family of integers (those representable as repetitions of a binary block) the minimal order of a graph with exactly n maximal independent sets is asymptotically equal to log₂ n. The authors achieve this by introducing the block graph construction, proving its correctness, extending it to arbitrary n via a decomposition strategy, and validating the improvement experimentally. This work closes a long‑standing gap between lower and upper bounds for the inverse MIS problem and opens a pathway for analogous inverse investigations across graph theory.