Number Of Maximal Independent Sets Research Articles

On the Maximum Number of Maximum Independent Sets of Bipartite Graphs

On the Maximum Number of Maximum Independent Sets of Bipartite Graphs

Counting maximal independent sets in some [formula omitted]-gonal cacti

Counting the number of maximal independent sets of graphs was started over 50 years ago by Erdős and Mooser. The problem has been continuously studied with a number of variations. Interestingly, when the maximal condition of an independent set is removed, such the concept presents one of topological indices in molecular graphs, the so called Merrifield–Simmons index. In this paper, we applied the concept of bivariate generating function to establish the recurrence relations of the numbers of maximal independent sets of regular n-gonal cacti when 3≤n≤6. By the ideas on meromorphic functions and the growth of power series coefficients, the asymptotic behaviors through simple functions of these recurrence relations have been established.

On the number of maximal independent sets: From Moon–Moser to Hujter–Tuza

AbstractWe connect two classical results in extremal graph theory concerning the number of maximal independent sets. The maximum number of maximal independent sets in an ‐vertex graph was determined by Miller and Muller and independently by Moon and Moser. The maximum number of maximal independent sets in an ‐vertex triangle‐free graph was determined by Hujter and Tuza. We give a common generalization of these results by determining the maximum number of maximal independent sets in an ‐vertex graph containing no induced triangle matching of size . This also improves a stability result of Kahn and Park on . Our second result is a new (short) proof of a second stability result of Kahn and Park on the maximum number of maximal independent sets in ‐vertex triangle‐free graphs containing no induced matching of size .

Spanning k-Ended Tree in 2-Connected Graph

Win proved a very famous conclusion that states the graph G with connectivity κ(G), independence number α(G) and α(G)≤κ(G)+k−1(k≥2) contains a spanning k-ended tree. This means that there exists a spanning tree with at most k leaves. In this paper, we strengthen the Win theorem to the following: Let G be a simple 2-connected graph such that |V(G)|≥2κ(G)+k, α(G)≤κ(G)+k(k≥2) and the number of maximum independent sets of cardinality κ+k is at most n−2κ−k+1. Then, either G contains a spanning k-ended tree or a subgraph of Kκ∨((k+κ−1)K1∪Kn−2κ−k+1).

Maximal independent sets in clique-free graphs

Nielsen proved that the maximum number of maximal independent sets (MISs) of size k in an n-vertex graph is asymptotic to (n/k)k, with the extremal construction a disjoint union of k cliques with sizes as close to n/k as possible. In this paper we study how many MISs of size k an n-vertex graph G can have if G does not contain a clique Kt. We prove for all fixed k and t that there exist such graphs with n⌊(t−2)kt−1⌋−o(1) MISs of size k by utilizing recent work of Gowers and B. Janzer on a generalization of the Ruzsa–Szemerédi problem. We prove that this bound is essentially best possible for triangle-free graphs when k≤4.

The number of maximal independent sets in trees with a given number of leaves

For any n and l, we completely describe trees with the maximum and minimum numbers of maximal independent sets among all n-vertex trees with l leaves.

The Number of Maximal Independent Sets in the Hamming Cube

Let Qn be the n-dimensional Hamming cube and N = 2n. We prove that the number of maximal independent sets in Qn is asymptotically $$2n{2^{N/4}},$$ as was conjectured by Ilinca and the first author in connection with a question of Duffus, Frankl and Rödl.The value is a natural lower bound derived from a connection between maximal independent sets and induced matchings. The proof that it is also an upper bound draws on various tools, among them “stability” results for maximal independent set counts and old and new results on isoperimetric behavior in Qn.

Trees with extremal numbers of k-dominating sets

In this paper we investigate the relation between the number of k-dominating sets and the number of independent sets in a tree. We call the k-interior of a tree T the forest that contains all the vertices of T of degree at least k. Heuberger and Wagner (2008) characterized for all n,d≥3 the n-vertex trees Td,n with maximum vertex degree d and the maximum number of independent sets. We show that for all n,k≥2, if a tree T has the maximum number of k-dominating sets among all n-vertex trees, then either it contains exactly 2⌊n−2k−1⌋k-dominating sets or its k-interior is isomorphic to a forest aK1∪Tk,b for some integers a,b≥0. Moreover, we completely describe trees with the minimum possible number of k-dominating sets for every k≥2.

The number of maximal independent sets of sequentially Cohen-Macaulay bipartite graphs

We determine the maximum number of maximal independent sets of sequentially bipartite graphs and we give a completed characterization of the extremal graphs.

Maximizing the number of independent sets of fixed size in Kn‐covered graphs

AbstractFor some given graph , a graph is called ‐covered if each vertex in is contained in a copy of . In this note, we determine the maximum number of independent sets of size in ‐vertex ‐covered graphs and classify the extremal graphs. The result answers a question proposed by Chakraborti and Loh. The proof uses an edge‐switching operation on hypergraphs which never increases the number of independent sets.

Maximal independent sets and regularity of graphs

In this paper, we give a lower bound of the number of maximal independent sets in a graph [Formula: see text] in terms of the Castelnuovo–Mumford regularity of its edge ideal. We also find two classes of graphs achieving this minimum value.

Vector weighted Stirling numbers and an application in graph theory

We introduce \textit{vector weighted Stirling numbers}, which are a generalization of ordinary Stirling numbers and restricted Stirling numbers. Some relations between vector weighted Stirling numbers and ordinary Stirling numbers and some of their applications are stated. Moreover, as an application of vector weighted Stirling numbers of the second kind in graph theory, we compute the number of maximal independent sets of different sizes in k -intersection graphs.

Trees with a given number of leaves and the maximal number of maximum independent sets

Abstract A complete description of trees with maximal possible number of maximum independent sets among all n-vertex trees with exactly l leaves is obtained. For all values of the parameters n and l the extremal tree is unique and is the result of merging the endpoints of l simple paths.

Some Combinatorial Problems in Power-Law Graphs

Abstract The power-law behavior is ubiquitous in a majority of real-world networks, and it was shown to have a strong effect on various combinatorial, structural and dynamical properties of graphs. For example, it has been shown that in real-life power-law networks, both the matching number and the domination number are relatively smaller, compared with homogeneous graphs. In this paper, we study analytically several combinatorial problems for two power-law graphs with the same number of vertices, edges and the same power exponent. For both graphs, we determine exactly or recursively their matching number, independence number, domination number, the number of maximum matchings, the number of maximum independent sets and the number of minimum dominating sets. We show that power-law behavior itself cannot characterize the combinatorial properties of a heterogenous graph. Since the combinatorial properties studied here have found wide applications in different fields, such as structural controllability of complex networks, our work offers insight in the applications of these combinatorial problems in power-law graphs.

The second largest number of maximum independent sets in trees of odd order without duplicated leaves

The second largest number of maximum independent sets in trees of odd order without duplicated leaves

From Independent Sets and Vertex Colorings to Isotropic Spaces and Isotropic Decompositions: Another Bridge between Graphs and Alternating Matrix Spaces

In the 1970s, Lovász built a bridge between graphs and alternating matrix spaces, in the context of perfect matchings [Proceedings of FCT, 1979, pp. 565--574]. A similar connection between bipartite graphs and matrix spaces plays a key role in the recent resolutions of the noncommutative rank problem [A. Garg et al., Proceedings of FOCS, 2016, pp. 109--117; G. Ivanyos, Y. Qiao, and K. V. Subrahmanyam, Comput. Complexity, 26 (2017), pp. 717--763]. In this paper, we lay the foundation for another bridge between graphs and alternating matrix spaces, in the context of independent sets and vertex colorings. The corresponding structures in alternating matrix spaces are isotropic spaces and isotropic decompositions, both useful structures in group theory and manifold theory. We first show that the maximum independent set problem and the vertex $c$-coloring problem reduce to the maximum isotropic space problem and the isotropic $c$-decomposition problem, respectively. Next, we show that several topics and results about independent sets and vertex colorings have natural correspondences for isotropic spaces and decompositions. These include algorithmic problems, such as the maximum independent set problem for bipartite graphs, and exact exponential-time algorithms for the chromatic number, as well as mathematical questions, such as the number of maximal independent sets, and the relation between the maximum degree and the chromatic number. These connections lead to new interactions between graph theory and algebra. Some results have concrete applications to group theory and manifold theory, and we initiate a variant of these structures in the context of quantum information theory. Finally, we propose several open questions for further exploration.

On the maximum number of maximum independent sets in connected graphs

AbstractWe characterize the connected graphs of given order and given independence number that maximize the number of maximum independent sets. For , there is a unique such graph that arises from the disjoint union of cliques of orders and , which is the complement of a Turán graph, by selecting a vertex in a largest clique and adding an edge between and a vertex in each of the remaining cliques. Our result confirms a conjecture of Derikvand and Oboudi.

The maximum number of maximum dissociation sets in trees

AbstractA subset of vertices is a maximum independent set if no two of the vertices are adjacent and the subset has maximum cardinality. A subset of vertices is called a maximum dissociation set if it induces a subgraph with vertex degree at most 1, and the subset has maximum cardinality. Zito proved that the maximum number of maximum independent sets of a tree of order is if is odd, and if is even and also characterized all extremal trees with the most maximum independent sets, which solved a question posed by Wilf. Inspired by the results of Zito, in this paper, by establishing four structure theorems and a result of ‐König–Egerváry graph, we show that the maximum number of maximum dissociation sets in a tree of order is and also give complete structural descriptions of all extremal trees on which these maxima are achieved.

An isoperimetric inequality for the Hamming cube and some consequences

Our basic result, an isoperimetric inequality for Hamming cube $Q_n$, can be written: \[ \int h_A^\beta d\mu \ge 2 \mu(A)(1-\mu(A)). \] Here $\mu$ is uniform measure on $V=\{0,1\}^n$ ($=V(Q_n)$); $\beta=\log_2(3/2)$; and, for $S\subseteq V$ and $x\in V$, \[ h_S(x) = \begin{cases} d_{V \setminus S}(x) &\mbox{ if } x \in S, 0 &\mbox{ if } x \notin S \end{cases} \] (where $d_T(x)$ is the number of neighbors of $x$ in $T$). This implies inequalities involving mixtures of edge and vertex boundaries, with related stability results, and suggests some more general possibilities. One application, a stability result for the set of edges connecting two disjoint subsets of $V$ of size roughly $|V|/2$, is a key step in showing that the number of maximal independent sets in $Q_n$ is $(1+o(1))2n\exp_2[2^{n-2}]$. This asymptotic statement, whose proof will appear separately, was the original motivation for the present work.

Null decomposition of bipartite graphs without cycles of length 0 modulo 4

In this work we study the null space of bipartite graphs without cycles of length 0 modulo 4 (denoted as C4k-free bipartite graphs), and its relation to structural properties. We extend the Null Decomposition of trees, introduced by Jaume and Molina (2018), to C4k-free bipartite graphs. This decomposition uses the null space of the adjacency matrix of a graph G to decompose it into two different types of graphs: CN(G) and CS(G). CN has perfect matching number. CS(G) has a unique maximum independent set. We obtain formulas for the independence number and the matching number of a C4k-free bipartite graph using this decomposition. We also show how the number of maximum matchings and the number of maximum independent sets in a C4k-free bipartite graph are related to its null decomposition.