Forbidden subgraphs and the König–Egerváry property

The matching number of a graph is the maximum size of a set of vertex-disjoint edges. The transversal number is the minimum number of vertices needed to meet every edge. A graph has the König–Egerváry property  if its matching number equals its transversal number. Lovász proved a characterization of graphs having the König–Egerváry property by means of forbidden subgraphs within graphs with a perfect matching. Korach, Nguyen, and Peis proposed an extension of Lovász’s result to a characterization of all graphs having the König–Egerváry property in terms of forbidden configurations (which are certain arrangements of a subgraph and a maximum matching). In this work, we prove a characterization of graphs having the König–Egerváry property by means of forbidden subgraphs which is a strengthened version of the characterization by Korach et al. Using our characterization of graphs with the König–Egerváry property, we also prove a forbidden subgraph characterization for the class of edge-perfect graphs.     

The number of independent sets in unicyclic graphs with a given diameter

Let U_n,d denote the set of unicyclic graphs with a given diameter d. In this paper, the unique unicyclic graph in U_n,d with the maximum number of independent sets, is characterized.     

Closed formulas for the numbers of small independent sets and matchings and an extremal problem for trees

We derive closed formulas for the numbers of independent sets of size at most 4 and matchings of size at most 3 in graphs without small cycles that depend only on the degree sequence and the products of the degrees of adjacent vertices.

As a related problem we describe an algorithm that determines a tree of given degree sequence that maximizes the sum of the products of the degrees of adjacent vertices.     

Maximal independent sets on a grid graph

An independent vertex set of a graph is a set of vertices of the graph in which no two vertices are adjacent, and a maximal independent set is one that is not a proper subset of any other independent set. In this paper we count the number of maximal independent sets of vertices on a complete rectangular grid graph. More precisely, we provide a recursive matrix-relation producing the partition function with respect to the number of vertices. The asymptotic behavior of the maximal hard square entropy constant is also provided. We adapt the state matrix recursion algorithm, recently invented by the author to answer various two-dimensional regular lattice model problems in enumerative combinatorics and statistical mechanics.     

Measure preserving homomorphisms and independent sets in tensor graph powers

In this note, we study the behavior of independent sets of maximum probability measure in tensor graph powers. To do this, we introduce an upper bound using measure preserving homomorphisms. This work extends some previous results concerning independence ratios of tensor graph powers.     

Two problems on independent sets in graphs

Let i_t(G) denote the number of independent sets of size t in a graph G. Levit and Mandrescu have conjectured that for all bipartite G the sequence (i_t(G))_t≥0 (the independent set sequence of G) is unimodal. We provide evidence for this conjecture by showing that this is true for almost all equibipartite graphs. Specifically, we consider the random equibipartite graph G(n,n,p), and show that for any fixed p∈(0,1] its independent set sequence is almost surely unimodal, and moreover almost surely log-concave except perhaps for a vanishingly small initial segment of the sequence. We obtain similar results for .We also consider the problem of estimating i(G)=∑_t≥0i_t(G) for G in various families. We give a sharp upper bound on the number of independent sets in an n-vertex graph with minimum degree δ, for all fixed δ and sufficiently large n. Specifically, we show that the maximum is achieved uniquely by K_δ,n−δ, the complete bipartite graph with δ vertices in one partition class and n−δ in the other.We also present a weighted generalization: for all fixed x>0 and δ>0, as long as n=n(x,δ) is large enough, if G is a graph on n vertices with minimum degree δ then ∑_t≥0i_t(G)x^t≤∑_t≥0i_t(K_δ,n−δ)x^t with equality if and only if G=K_δ,n−δ.     

The extreme set condition of a graph

Let G be a simple graph. The size of any largest matching in G is called the matching number of G and is denoted by ν(G). Define the deficiency of G, def(G), by the equation def(G)=|V(G)|−2ν(G). A set of points X in G is called an extreme set if def(G−X)=def(G)+|X|. Let c₀(G) denote the number of the odd components of G. A set of points X in G is called a barrier if c₀(G−X)=def(G)+|X|. In this paper, we obtain the following:

(1) Let G be a simple graph containing an independent set of size i, where i2. If X is extreme in G for every independent set X of size i in G, then there exists a perfect matching in G.

(2) Let G be a connected simple graph containing an independent set of size i, where i2. Then X is extreme in G for every independent set X of size i in G if and only if G=(U,W) is a bipartite graph with |U|=|W|i, and |Γ(Y)||U|−i+m+1 for any Y U, |Y|=m (1mi−1).

(3) Let G be a connected simple graph containing an independent set of size i, where i2. Then X is a barrier in G for every independent set X of size i in G if and only if G=(U,W) is a bipartite graph with |U|=|W|=i, and |Γ(Y)|m+1 for any Y U, |Y|=m (1mi−1).     

First eigenvalue and first eigenvectors of a nonsingular unicyclic mixed graph

Let G be a mixed graph and let L(G) be the Laplacian matrix of the graph G. The first eigenvalue and the first eigenvectors of G are respectively referred to the least nonzero eigenvalue and the corresponding eigenvectors of L(G). In this paper we focus on the properties of the first eigenvalue and the first eigenvectors of a nonsingular unicyclic mixed graph (abbreviated to a NUM graph). We introduce the notion of characteristic set associated with the first eigenvectors, and then obtain some results on the sign structure of the first eigenvectors. By these results we determine the unique graph which minimizes the first eigenvalue over all NUM graphs of fixed order and fixed girth, and the unique graph which minimizes the first eigenvalue over all NUM graphs of fixed order.     

Partition the vertices of a graph into one independent set and one acyclic set

For a given graph G, if the vertices of G can be partitioned into an independent set and an acyclic set, then we call G a near-bipartite graph. This paper studies the recognition of near-bipartite graphs. We give simple characterizations for those near-bipartite graphs having maximum degree at most 3 and those having diameter 2. We also show that the recognition of near-bipartite graphs is NP-complete even for graphs where the maximum degree is 4 or where the diameter is 4.     

On minimum rank and zero forcing sets of a graph

For a graph G on n vertices and a field F, the minimum rank of G over F, written as mr^F(G), is the smallest possible rank over all n×n symmetric matrices over F whose (i,j)th entry (for ) is nonzero whenever ij is an edge in G and is zero otherwise. The maximum nullity of G over F is M^F(G)=n-mr^F(G). The minimum rank problem of a graph G is to determine mr^F(G) (or equivalently, M^F(G)). This problem has received considerable attention over the years. In [F. Barioli, W. Barrett, S. Butler, S.M. Cioab?, D. Cvetkovi?, S.M. Fallat, C. Godsil, W. Haemers, L. Hogben, R. Mikkelson, S. Narayan, O. Pryporova, I. Sciriha, W. So, D. Stevanovi?, H. van der Holst, K.V. Meulen, A.W. Wehe, AIM Minimum Rank-Special Graphs Work Group, Zero forcing sets and the minimum rank of graphs, Linear Algebra Appl. 428 (2008) 1628-1648], a new graph parameter Z(G), the zero forcing number, was introduced to bound M^F(G) from above. The authors posted an attractive question: What is the class of graphs G for which Z(G)=M^F(G) for some field F? This paper focuses on exploring the above question.     

On scrambled sets and a theorem of Kuratowski on independent sets

The measure of scrambled sets of interval self-maps was studied by many authors, including Smítal, Misiurewicz, Bruckner and Hu, and Xiong and Yang. In this note, first we introduce the notion of ``-chaos" which is related to chaos in the sense of Li-Yorke, and we prove a general theorem which is an improvement of a theorem of Kuratowski on independent sets. Second, we apply the result to scrambled sets of higher dimensional cases. In particular, we show that if a map of the unit -cube is -chaotic on , then for any there is a map such that and are topologically conjugate, and has a scrambled set which has Lebesgue measure 1, and hence if , then there is a homeomorphism with a scrambled set satisfying that is an -set in and has Lebesgue measure 1.

     

NEIGHBORHOOD UNION OF INDEPENDENT SETS AND HAMILTONICITY OF CLAW-FREE GRAPHS

Let G be a graph,for any u∈V(G),let N(u) denote the neighborhood of u and d(u)=|N(u)| be the degree of u. For any U V(G) ,let N(U)=Uu,∈UN(u), and d(U)=|N(U)|.A graph G is called claw-free if it has no induced subgraph isomorphic to K1.3. One of the fundamental results concerning cycles in claw-free graphs is due to Tian Feng,et al. : Let G be a 2-connected claw-free graph of order n,and d(u) d(v) d(w)≥n-2 for every independent vertex set {u,v,w} of G, then G is Hamiltonian. It is proved that, for any three positive integers s ,t and w,such that if G is a (s t w-1)connected claw-free graph of order n,and d(S) d(T) d(W)＞n-(s t w) for every three disjoint independent vertex sets S,T,W with |S |=s, |T|=t, |W|=w,and S∪T∪W is also independent ,then G is Hamiltonian. Other related results are obtained too.     

Measures on independent sets, a quantitative version of Rudin's theorem

We construct measures with independent support whose Fourier coefficients decrease as fast as possible.

     

A simple proof of an inequality connecting the alternating number of independent sets and the decycling number

If s_k denotes the number of independent sets of cardinality k and α(G) is the size of a maximum independent set in graph G, then I(G;x)=s₀+s₁x+?+s_α(G)x^α(G) is the independence polynomial of G (Gutman and Harary, 1983) [8].In this paper we provide an elementary proof of the inequality|I(G;−1)|≤2^φ(G) (Engström, 2009) [7], where φ(G) is the decycling number of G (Beineke and Vandell, 1997) [3], namely, the minimum number of vertices that have to be deleted in order to turn G into a forest.     

Invariant Gaussian processes and independent sets on regular graphs of large girth

We prove that every 3‐regular, n‐vertex simple graph with sufficiently large girth contains an independent set of size at least 0.4361n. (The best known bound is 0.4352n.) In fact, computer simulation suggests that the bound our method provides is about 0.438n. Our method uses invariant Gaussian processes on the d‐regular tree that satisfy the eigenvector equation at each vertex for a certain eigenvalue . We show that such processes can be approximated by i.i.d. factors provided that . We then use these approximations for to produce factor of i.i.d. independent sets on regular trees. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 284–303, 2015     

Counting the number of independent sets in chordal graphs

We study some counting and enumeration problems for chordal graphs, especially concerning independent sets. We first provide the following efficient algorithms for a chordal graph: (1) a linear-time algorithm for counting the number of independent sets; (2) a linear-time algorithm for counting the number of maximum independent sets; (3) a polynomial-time algorithm for counting the number of independent sets of a fixed size. With similar ideas, we show that enumeration (namely, listing) of the independent sets, the maximum independent sets, and the independent sets of a fixed size in a chordal graph can be done in constant time per output. On the other hand, we prove that the following problems for a chordal graph are #P-complete: (1) counting the number of maximal independent sets; (2) counting the number of minimum maximal independent sets. With similar ideas, we also show that finding a minimum weighted maximal independent set in a chordal graph is NP-hard, and even hard to approximate.     

Some remarks on the geodetic number of a graph

A set of vertices D of a graph G is geodetic if every vertex of G lies on a shortest path between two not necessarily distinct vertices in D. The geodetic number of G is the minimum cardinality of a geodetic set of G.We prove that it is NP-complete to decide for a given chordal or chordal bipartite graph G and a given integer k whether G has a geodetic set of cardinality at most k. Furthermore, we prove an upper bound on the geodetic number of graphs without short cycles and study the geodetic number of cographs, split graphs, and unit interval graphs.     

Matchings and independent sets of a fixed size in regular graphs

We use an entropy based method to study two graph maximization problems. We upper bound the number of matchings of fixed size ℓ in a d-regular graph on N vertices. For bounded away from 0 and 1, the logarithm of the bound we obtain agrees in its leading term with the logarithm of the number of matchings of size ℓ in the graph consisting of disjoint copies of K_d,d. This provides asymptotic evidence for a conjecture of S. Friedland et al. We also obtain an analogous result for independent sets of a fixed size in regular graphs, giving asymptotic evidence for a conjecture of J. Kahn. Our bounds on the number of matchings and independent sets of a fixed size are derived from bounds on the partition function (or generating polynomial) for matchings and independent sets.