We characterize the graphs which are simultaneously α-critical and members of the class 𝐖_p. The characterization is stated in three equivalent languages. In the graph itself, such a graph is a well-covered graph whose codimension-one localization fibers all have size at least p and whose edges are exactly covered by the cliques induced by those fibers. In the independence complex, it is a pure flag complex in which every ridge has degree at least p and every missing edge is generated by the link of a ridge. In the complement, it is a K_r+1-saturated graph, where r=α(G), all maximal cliques have size r, and the minimum (r-1)-clique-codegree is at least p. This gives an exact formula for the largest p for which a well-covered graph belongs to 𝐖_p. We make this complement correspondence explicit, record saturation-theoretic consequences including dense-complement rigidity and p-sensitive edge and order bounds, and give a family of sharp examples showing that the local sufficient condition from the recent work of Hoang, Levit and Mandrescu is not necessary outside the locally triangle-free setting, for all p≥2.

Let $\alpha(G)$ denote the cardinality of a maximum independent set, while $\mu(G)$ be the size of a maximum matching in $G=\left( V,E\right) $. It is known that if $\alpha(G)+\mu(G)=\left\vert V\right\vert $, then $G$ is a \textit{König-Egerváry graph. If $\alpha (G)+\mu(G)=\left\vert V\right\vert -1$, then $G$ is an $1$-König-Egerváry graph. If $G$ is not a König-Egerváry graph, and there exists a vertex $v\in V$ (an edge $e\in E$) such that $G-v$ ($G-e$) is König-Egerváry, then $G$ is called a vertex (an edge) almost König-Egerváry graph (respectively). In this paper, we characterize all these types of almost König-Egerváry graphs and present interrelationships between them.

It was proved in (Levit and Mandrescu, 2022) that both (V(G), Crown(G)) and (V(G), CritIndep(G)) are augmentoids, established partial augmentation phenomena for the family Ψ(G) of local maximum independent sets, and asked in Problem 5.5 to characterize the graphs whose family Ψ(G) is an augmentoid. We prove that the answer is positive in full generality: for every finite simple graph G, the set system (V(G),Ψ(G)) is an augmentoid. The proof is constructive. If S,T(G), then the explicit choice A=S ∖ N[T], B=T ∖ N[S] satisfies T∪ A(G), S∪ B(G), |T∪ A|=|S∪ B|. As a structural consequence, for every fixed S(G) the map T↦ S∪ T induces a canonical bijection from Ψ(G-N[S]) onto the members of Ψ(G) containing S, and α(G)=|S|+α(G-N[S]). This decomposition also yields explicit formulas for the intersection and the union of all the maximum independent sets extending S, together with counting formulas for the local maximum and maximum independent sets containing S. We also add a short visual guide to the framework CritIndep(G) ⊆ Crown(G)⊆ Psi(G) and end with several natural follow-up problems suggested by the theorem.

We establish new characterizations of graphs belonging to the Wp class. In addition, we characterize locally triangle-free alpha-critical graphs in this class. As a consequence, our results yield a partial answer to a question raised by Plummer [M.D. Plummer, Well-covered graphs: A survey, Quaest. Math. 16 (1993) 253-287] in the case p = 2.

Let G be a finite simple graph. An independent set I of G is critical if |I|-|N(I)|≥|J|-|N(J)| for every independent set J of G. A critical independent set is maximum if it has maximum cardinality. The core and the nucleus of G are defined as the intersection of all maximum independent sets and the intersection of all maximum critical independent sets, respectively. In 2019, Jarden, Levit, and Mandrescu posed the problem of characterizing the graphs satisfying core(G)=nucleus(G). In this paper, we provide a complete solution to this problem. Using Larson's independence decomposition, which partitions any graph into a König–Egerváry component L_G an a 2-bicritical component L_G^c, we establish that core(G)=nucleus(G) holds if and only if core (L_G^c)=∅ and no vertex of corona(G) lies in the boundary between L_G and L_G^c. We also show that the same boundary condition is equivalent to the identity diadem(G)=corona(G) ∩ L(G). Several consequences and related structural properties are also derived.

For a positive integer p , a graph $G$ belongs to class W p if |V(G)| > p and any p disjoint independent sets are contained in p disjoint maximum independent sets. In this paper, we establish some inequalities for the coefficients of the independence polynomial of a W p graph.It is then proven that the clique corona graph G ο K p of a graph G and a complete graph K p belongs to W p . As an application, weexamine the unimodality problem for the independence polynomial of G ο K p .In addition we prove the unimodality of the independence polynomial ofS n ο K p , where S n is a star graph.

Let G be a graph of order n. For a positive integer p, G is said to be a Wp graph if n >= p and every p pairwise disjoint independent sets of G are contained within p pairwise disjoint maximum independent sets. In this paper, we establish that every connected Wp graph G is p-quasi-regularizable if and only if n >= (p + 1) & centerdot; alpha, where alpha is the independence number of G and p =/ 2. This finding ensures that the independence polynomial of a connected Wp graph G is log-concave whenever (p + 1) & centerdot; alpha <= n <= p & centerdot; alpha + 2 root p & centerdot; alpha + p and alpha 2 <= p, or p & centerdot; alpha + 2 root p & centerdot; alpha + p 2+1)& centerdot;p+(alpha-1)2 alpha+1 <= p. Moreover, the clique corona graph G degrees Kp serves as an example of the Wp graph class. We further demonstrate that the independence polynomial of G degrees Kpis always log-concave for sufficiently large p. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

A graph is said to be K & odblac;nig-Egerv & aacute;ry if its matching number equals its vertex cover number. The difference between these two graph parameters, the vertex cover number minus the matching number, measures, in some sense, how far a graph is from being a K & odblac;nig-Egerv & aacute;ry graph. Several properties of this difference, called the K & odblac;nig- Egerv & aacute;ry index or K & odblac;nig deficiency, are presented, including some nontrivial structural characterizations. Furthermore, it is shown that various statements involving K & odblac;nig- Egerv & aacute;ry graphs are, in fact, general statements about graphs that can be expressed in terms of their K & odblac;nig-Egerv & aacute;ry indices. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

A set S subset of V is independent in a graph G = (V, E) if no two vertices from S are adjacent. The independence number alpha(G) is the cardinality of a maximum independent set, while mu(G) is the size of a maximum matching in G. If alpha(G)+mu(G) equals the order of G, then G is a K & ouml;nig-Egerv & aacute;ry graph (Deming, 1979; Gavril, 1977; Sterboul, 1979). The number d (G) = max{eAe - eN (A)e : A subset of V} is the critical difference of G (Zhang, 1990) (where N (A) = {v : v E V, N (v) f1A not equal empty set}). It is known that the inequality alpha(G) - mu(G) <= d (G) holds for every graph (Levit and Mandrescu, 2012; Lorentzen, 1966; Schrijver, 2003). one odd cycle. Let ker(G) = boolean AND{S : S is a critical independent set of G}, core(G) be A graph G is (i) unicyclic if it has a unique cycle, (ii) almost bipartite if it has only the intersection of all maximum independent sets, and corona(G) be the union of all maximum independent sets of G. It is known that ker(G) subset of core(G) for every graph (Levit and Mandrescu, 2012), while the equality holds for bipartite graphs (Levit and Mandrescu, 2013), and for unicyclic non-K & ouml;nig-Egerv & aacute;ry graphs (Levit and Mandrescu, 2014). In this paper, we prove that if G is an almost bipartite non-K & ouml;nig-Egerv & aacute;ry graph, then ker(G) = core(G), corona(G) boolean OR N(core(G)) = V(G), and ecorona(G)e + ecore(G)e = 2 alpha(G) + 1. (c) 2025 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

Let α (G) denote the cardinality of a maximum independent set and μ (G) be the size of a maximum matching of a graph G=( V( G) ,E( G) ) . If α (G)+μ (G)=| V( G) | -k , then G is a k -König–Egerváry graph. In particular, if k=0 , then G is a König–Egerváry graph. The corona H∘𝒳 of a graph H and a family of graphs 𝒳={ X_i:1≤ i≤| V(H)| } is obtained by joining each vertex v_i of H to all the vertices of the corresponding graph X_i,i=1,2,...,| V(H)| . In this paper we completely characterize graphs whose coronas are k-König–Egerváry graphs, where k∈{ 0,1} .

Let α(G) and μ(G) denote the cardinality of a maximum independent set and the size of a maximum matching, respectively, in the graph G= (V,E) . If α(G)+μ(G)= | V | , then G is a Kőnig–Egerváry graph. The number d (G) =max{| A | - | N (A) | :A⊆ V} is the critical difference of the graph G, where N (A) ={ v:v∈ V,N (v) ∩ A≠∅} . Every set B⊆ V satisfying d (G) = | B | - | N (B) | is critical. Let ε (G) = |(G) | and ξ (G) = |core (G) | , where (G) is the intersection of all critical independent sets, and core (G) is the intersection of all maximum independent sets. It is known that (G)⊆ core (G) holds for every graph. Let us define Clearly, ϱ_v (G) = | V | and ϱ_e (G) = | E | for bipartite graphs. Unlike the bipartiteness, the property of being a Kőnig–Egerváry graph is not hereditary. In this paper, we show that ϱ_v (G) = | V | -ξ (G) +ε (G)aaandaaϱ_e (G) ≥| E | -ξ (G) +ε (G) for every Kőnig–Egerváry graph G.

Let α(G) denote the cardinality of a maximum independent set and μ(G) be the size of a maximum matching of a graph G=( V,E). If α(G)+μ(G)=| V|, then G is a König-Egerváry graph, and G is a 1-König-Egerváry graph whenever α(G)+μ(G)=| V| -1. The corona H∘𝒳 of a graph H and a family of graphs 𝒳={ X_i:1≤ i≤| V(H)|} is obtained by joining each vertex v_i of H to all the vertices of the corresponding graph X_i,i=1,2,...,| V(H)|. In this paper we completely characterize graphs whose coronas are k-König-Egerváry graphs, where k∈{ 0,1}.

In this paper, our research primarily focuses on f-symmetric and unimodal properties of independence polynomials I(G∘ (K_p∪ K_q);x) . More precisely, we prove that the independence polynomial I(G∘ (K_p∪ K_q);x) is unimodal for every graph G, whenever the positive integers p and q are large enough. In addition, we establish several inequalities involving the coefficients of I(G∘ (K_p∪ K_q);x) for arbitrary p and q.

If for any $k$ the $k$-th coefficient of a polynomial $I(G;x)$ is equal to the number of stable sets of cardinality $k$ in the graph $G$, then it is called the independence polynomial of $G$ (Gutman and Harary, 1983). Alavi, Malde, Schwenk and Erdos (1987) conjectured that $I(G;x)$ is unimodal, whenever $G$ is a forest, while Brown, Dilcher and Nowakowski (2000) conjectured that $I(G;x)$ is unimodal for any well-covered graph G. Michael and Traves (2003) showed that the assertion is false for well-covered graphs with $a(G)$ > 3 ($a(G)$ is the size of a maximum stable set of the graph $G$), while for very well-covered graphs the conjecture is still open. In this paper we give support to both conjectures by demonstrating that if $a(G)$ < 4, or $G$ belongs to ${K_{1,n}, P_{n}: n > 0}$, then $I(G*;x)$ is log-concave, and, hence, unimodal (where $G*$ is the very well-covered graph obtained from $G$ by appending a single pendant edge to each vertex).

In this paper, our research primarily focuses on f -symmetric and unimodal properties of independence polynomials I (G circle (K-p boolean OR K-q); x). More precisely, we prove that the independence polynomial I (G circle (K-p boolean OR K-q)); x) is unimodal for every graph G, whenever the positive integers p and q are large enough. In addition, we establish several inequalities involving the coefficients of I (G circle (K-p boolean OR K-q)); x) for arbitrary p and q.

If $$\alpha (G)+\mu (G)=\left| V\right|$$ , then $$G=\left( V,E\right)$$ is a König–Egerváry graph, where $$\alpha (G)$$ denotes the cardinality of a maximum independent set, while $$\mu (G)$$ is the size of a maximum matching in G. If $$d_{1}\le d_{2}\le \cdots \le d_{n}$$ is the degree sequence of G, then the annihilation number $$a\left( G\right)$$ of G is the largest integer k such that $$\sum \limits _{i=1} ^{k}d_{i}\le \left| E\right|$$ (Pepper, Binding independence, Ph.D. Dissertation, University of Houston, 2004; Pepper, On the annihilation number of a graph, in: Recent Advances in Electrical Engineering: Proceedings of the 15th American Conference on Applied Mathematics, pp 217–220, 2009). A set $$A\subseteq V$$ satisfying $$\sum \limits _{a\in A}deg (a)\le \left| E\right|$$ is an annihilation set; if, in addition, $$deg \left( v\right) +\sum \limits _{a\in A}deg (a)>\left| E\right|$$ , for every vertex $$v\in V(G)-A$$ , then A is a maximal annihilation set in G. In Larson and Pepper (Graphs with equal independence and annihilation numbers. Electron J Comb 18:180, 2011) it was conjectured that the following assertions are equivalent: (i) $$\alpha \left( G\right) =a\left( G\right)$$ ; (ii) G is a König–Egerváry graph and every maximum independent set is a maximal annihilating set. Recently, it turned out that the implication “(i) $$\Longrightarrow$$ (ii)” was not true. A series of corresponding counterexamples can be found in Hiller (Counterexamples to the characterisation of graphs with equal independence and annihilation number. arXiv:2202.07529v1 [math.CO], 2022). In Levit and Mandrescu (On an annihilation number conjecture. Ars Math. Contemp. 18, 359–369, 2020), we presented an infinite family of non-bipartite König–Egerváry graphs that invalidate the “ (ii) $$\Longrightarrow$$ (i)” part of this conjecture. In this paper, we provide two more infinite families of counterexamples, one consisting of trees and the other one comprising non-tree bipartite graphs. We also show that the above conjecture is true for trees with $$\alpha \left( G\right) =4$$ , disconnected non-bipartite König–Egerváry graphs with $$\alpha \left( G\right) =4$$ , and disconnected bipartite graphs with $$\alpha \left( G\right) =4$$ excluding the three following counterexamples: $$C_{4}\cup 2K_{2},Domino\cup K_{2}$$ and $$K_{3,3}-e$$ .

An independent set A is maximal if it is not a proper subset of an independent set, while A is maximum if it has a maximum size. The problem of whether a graph has a pair of disjoint maximal independent sets was introduced by C. Berge in early 70's. The class of graphs for which every induced subgraph admits two disjoint maximal independent sets was characterized in (Shaudt, 2015). It is known that deciding whether a graph has two disjoint maximal independent sets is a NP-complete problem (Henning et al., 2009). In this paper, we are focused on finding conditions ensuring the existence of two disjoint maximum independent sets.

The independence number α(G) is the cardinality of a maximum independent set, while μ(G) is the size of a maximum matching in G. If α(G)+μ(G) equals the order of G, then G is called a Konig-Egervary graph. The number d( G) =max{| A| -| N( A) | :A⊆ V} is called the critical difference of G (where N( A) ={ v:v∈ V,N( v) ∩ A≠∅}). It is known that α(G)-μ(G)≤ d( G) holds for every graph. A graph G is unicyclic if it has a unique cycle and almost bipartite if it has only one odd cycle. Let (G)=⋂{ S:S is a critical independent set}, core( G) be the intersection of all maximum independent sets, and corona( G) be the union of all maximum independent sets of G. It is known that (G)⊆core(G) is true for every graph, while the equality holds for bipartite graphs, and for unicyclic non-Konig-Egervary graphs. In this paper, we prove that if G is an almost bipartite non-Konig-Egervary graph, then (G)= core(G), corona(G) ∪ N(core( G) )=V(G), and |corona(G)| +|core(G)| =2α(G)+1.