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An independent set in a graph is a collection of vertices that are not adjacent to each other. The cardinality of the largest independent set in $G$ is represented by $\alpha(G)$. The independence polynomial of a graph $G = (V, E)$ was introduced by Gutman and Harary in 1983 and is defined as \[ I(G;x) = \sum_{k=0}^{\alpha(G)}{s_k}x^{k}={s_0}+{s_1}x+{s_2}x^{2}+...+{s_{\alpha(G)}}x^{\alpha(G)}, \] where $s_k$ represents the number of independent sets in $G$ of size $k$. The conjecture made by Alavi, Malde, Schwenk, and Erd\"os in 1987 stated that the independence polynomials of trees are unimodal, and many researchers believed that this conjecture could be strengthened up to its corresponding log-concave version. However, in our paper, we present evidence that contradicts this assumption by introducing infinite families of trees whose independence polynomials are not log-concave., Comment: 24 pages, 9 figures

Let $G$ be a graph. A set $S \subseteq V(G)$ is independent if its elements are pairwise non-adjacent. A vertex $v \in V(G)$ is shedding if for every independent set $S \subseteq V(G) \setminus N[v]$ there exists $u \in N(v)$ such that $S \cup \{u\}$ is independent. An independent set $S$ is maximal if it is not contained in another independent set. An independent set $S$ is maximum if the size of every independent set of $G$ is not bigger than $|S|$. The size of a maximum independent set of $G$ is denoted $\alpha(G)$. A graph $G$ is well-covered if all its maximal independent sets are maximum, i.e. the size of every maximal independent set is $\alpha(G)$. The graph $G$ belongs to class $\mathbf{W_2}$ if every two pairwise disjoint independent sets in $G$ are included in two pairwise disjoint maximum independent sets. If a graph belongs to the class $\mathbf{W_2}$ then it is well-covered. Finding a maximum independent set in an input graph is an NP-complete problem. Recognizing well-covered graphs is co-NP-complete. The complexity status of deciding whether an input graph belongs to the $\mathbf{W_2}$ class is not known. Even when the input is restricted to well-covered graphs, the complexity status of recognizing graphs in $\mathbf{W_2}$ is not known. In this article, we investigate the connection between shedding vertices and $\mathbf{W_2}$ graphs. On the one hand, we prove that recognizing shedding vertices is co-NP-complete. On the other hand, we find polynomial solutions for restricted cases of the problem. We also supply polynomial characterizations of several families of $\mathbf{W_2}$ graphs., Comment: 15 pages, 2 figures, 1 table

Given a finite set $E$ and an operator $\sigma:2^{E}\longrightarrow2^{E}$, two sets $X,Y\subseteq E$ are \textit{cospanning} if $\sigma\left( X\right) =\sigma\left( Y\right) $. Corresponding \textit{cospanning equivalence relations} were investigated for greedoids in much detail (Korte, Lovasz, Schrader; 1991). For instance, these relations determine greedoids uniquely. In fact, the feasible sets of a greedoid are exactly the inclusion-wise minimal sets of the equivalence classes. In this research, we show that feasible sets of convex geometries are the inclusion-wise maximal sets of the equivalence classes of the corresponding closure operator. Same as greedoids, convex geometries are uniquely defined by the corresponding cospanning relations. For each closure operator $\sigma$, an element $x\in X$ is \textit{an extreme point} of $X$ if $x\notin\sigma(X-x)$. The set of extreme points of $X$ is denoted by $ex(X)$. We prove, that if $\sigma$ has the anti-exchange property, then for every set $X$ its equivalence class $[X]_{\sigma}$ is the interval $[ex(X),\sigma(X)]$. It results in the one-to-one correspondence between the cospanning partitions of an antimatroid and its complementary convex geometry. The obtained results are based on the connection between violator spaces, greedoids, and antimatroids. Cospanning characterization of these combinatorial structures allows us not only to give the new characterization of antimatroids and convex geometries but also to obtain the new properties of closure operators, extreme point operators, and their interconnections., Comment: 20 pages. arXiv admin note: text overlap with arXiv:2105.05074

Let alpha(G) denote the maximum size of an independent set of vertices and mu(G) be the cardinality of a maximum matching in a graph G. A matching saturating all the vertices is perfect. If alpha(G) + mu(G) equals the number of vertices of G, then it is called a Konig-Egervary graph. A graph is unicyclic if it has a unique cycle. In 2010, Bartha conjectured that a unique perfect matching, if it exists, can be found in O(m) time, where m is the number of edges. In this paper we validate this conjecture for Konig-Egervary graphs and unicylic graphs. We propose a variation of Karp-Sipser leaf-removal algorithm (Karp and Spiser, 1981), which ends with an empty graph if and only if the original graph is a Konig-Egervary graph with a unique perfect matching obtained as an output as well. We also show that a unicyclic non-bipartite graph G may have at most one perfect matching, and this is the case where G is a Konig-Egervary graph., Comment: 10 pages, 5 figures

We define two recursive functions obtained by decomposition of a given interval into four close parts and prove two lemmas which determine features of these functions., Comment: 18 pages

The paper investigates relationship between algebraic expressions and graphs. We consider a digraph called a square rhomboid that is an example of non-series-parallel graphs. Our intention is to simplify the expressions of square rhomboids and eventually find their shortest representations. With that end in view, we describe the new algorithm for generating square rhomboid expressions, which improves on our previous algorithms.

A graph is well-covered if every maximal independent set has the same cardinality. The recognition problem of well-covered graphs is known to be co-NP-complete. Let w be a weight function defined on the vertices of G. Then G is w-well-covered if all maximal independent sets of G are of the same weight. The set of weight functions w for which a graph is w-well-covered is a vector space. We prove that finding the vector space of weight functions under which an input graph is w-well-covered can be done in polynomial time, if the input graph does not contain cycles of length 4, 5, 6 and 7., Comment: 10 pages

The stability number of a graph G, denoted by alpha(G), is the cardinality of a maximum stable set, and mu(G) is the cardinality of a maximum matching in G. If alpha(G)+mu(G) equals its order, then G is a Konig-Egervary graph. In this paper we deal with square-stable graphs, i.e., the graphs G enjoying the equality alpha(G)=alpha(G^{2}), where G^{2} denotes the second power of G. In particular, we show that a Konig-Egervary graph is square-stable if and only if it has a perfect matching consisting of pendant edges, and in consequence, we deduce that well-covered trees are exactly the square-stable trees., Comment: 12 pages, 9 figures

A maximum stable set in a graph G is a stable set of maximum cardinality. S is a local maximum stable set of G, if S is a maximum stable set of the subgraph induced by its closed neighborhood. It is known that the family of all local maximum stable sets of a forest forms a greedoid on its vertex set. Bipartite, triangle-free, and well-covered graphs whose families of local maximum stable sets form greedoids have been analyzed as well. A unicycle graph owns only one cycle. In this paper we characterize the unicycle graphs whose families of local maximum stable sets form greedoids., Comment: 9 pages; 4 figures

If alpha=alpha(G) is the maximum size of an independent set and s_{k} equals the number of stable sets of cardinality k in graph G, then I(G;x)=s_{0}+s_{1}x+...+s_{alpha}x^{alpha} is the independence polynomial of G. In this paper we prove that: 1. I(T;-1) equels either -1 or 0 or 1 for every tree T; 2. I(G;-1)=0 for every connected well-covered graph G of girth > 5, non-isomorphic to C_{7} or K_{2}; 3. the absolute value of I(G;-1) is not greater than 2^nu(G), for every graph G, where nu(G) is its cyclomatic number., 16 pages; 13 figures

A maximum stable set in a graph G is a stable set of maximum cardinality. S is a local maximum stable set of G, if S is a maximum stable set of the subgraph induced by its closed neighborhood. Nemhauser and Trotter Jr. proved in 1975 that any local maximum stable set is a subset of a maximum stable set of G. In 2002 we showed that the family of all local maximum stable sets of a forest forms a greedoid on its vertex set. The cases where G is bipartite, triangle-free, well-covered, while the family of all local maximum stable sets is a greedoid, were analyzed in 2004, 2007, and 2008, respectively. In this paper we demonstrate that if the family of all local maximum stable sets of the graph satisfies the accessibility property, then it is an interval greedoid. We also characterize those graphs whose families of local maximum stable sets are either antimatroids or matroids., Comment: 13 pages, 11 figures

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 graph $G$, then it is called the independence polynomial of $G$ (Gutman and Harary, 1983). J. I. Brown, K. Dilcher and R. J. Nowakowski (2000) conjectured that the independence polynomial of a well-covered graph $G$ (i.e., a graph whose all maximal independent sets are of the same size) is unimodal, that is, there exists an index $k$ such that the part of the sequence of coefficients from the first to $k$-th is non-decreasing while the other part of coefficients is non-increasing. T. S. Michael and N. Traves (2002) provided examples of well-covered graphs whose independence polynomials are not unimodal. A. Finbow, B. Hartnell and R. J. Nowakowski (1993) proved that under certain conditions, any well-covered graph equals G* for some $G$, where G* is the graph obtained from $G$ by appending a single pendant edge to each vertex of $G$. Y. Alavi, P. J. Malde, A. J. Schwenk and P. Erd\"{o}s (1987) asked whether for trees the independence polynomial is unimodal. V. E. Levit and E. Mandrescu (2002) validated the unimodality of the independence polynomials of some well-covered trees (e.g., $P_{n}^{*},K_{1,n}^{*}$, where $P_{n}$ is the path on $n$ vertices and $K_{1,n}$ is the $n$-star graph). In this paper we show that for any graph $G$ with the stability number alpha(G) < 5, the independence polynomial of G* is unimodal., Comment: This paper has been presented at Thirty-fourth Southeastern International Conference on Combinatorics, Graph Theory, and Computing, March 2003, Florida Atlantic University, USA; 13 pages; 2 figures

The number of stable sets of cardinality $k$ in graph $G$ is the $k$-th coefficient of the independence polynomial of $G$ (I. Gutman and F. Harary, 1983). In 1990, Y. O. Hamidoune proved that for any claw-free graph, its independence polynomial is unimodal, i.e., there exists a coefficient $k$ such that the part of the sequence of coefficients from the first to $k$-th is non-decreasing while the second part of coefficients is non-increasing. Y. Alavi, P. J. Malde, A. J. Schwenk and P. Erd\"{o}s (1987) asked whether for trees (or perhaps forests) the independence polynomial is unimodal. J. I. Brown, K. Dilcher and R. J. Nowakowski (2000) conjectured that it is true for any well-covered graph (a graph whose all maximal independent sets have the same size). V. E. Levit and E. Mandrescu (1999) demonstrated that every well-covered tree can be obtained as a join of a number of well-covered spiders, where a spider is a tree having at most one vertex of degree at least three. In this paper we show that the independence polynomial of any well-covered spider is unimodal. In addition, we introduce some graph transformations respecting independence polynomials. They allow us to reduce several types of well-covered trees to claw-free graphs, and, consequently, to prove that their independence polynomials are unimodal., Comment: A preliminary version of this paper was presented at Second Haifa Workshop on Interdisciplinary Applications, of Graph Theory, Combinatorics and Algorithms June 17-20, 2002. 21 pages, 15 figures

The stability number of a graph G, denoted by alpha(G), is the cardinality of a stable set of maximum size in G. A graph is well-covered if every maximal stable set has the same size. G is a Koenig-Egervary graph if its order equals alpha(G) + mu(G), where mu(G) is the cardinality of a maximum matching in G. In this paper we characterize $��^{++}$-stable graphs, namely, the graphs whose stability numbers are invariant to adding any two edges from their complements. We show that a K��nig-Egerv��ry graph is $��^{++}$-stable if and only if it has a perfect matching consisting of pendant edges and no four vertices of the graph span a cycle. As a corollary it gives necessary and sufficient conditions for $��^{++}$-stability of bipartite graphs and trees. For instance, we prove that a bipartite graph is $��^{++}$-stable if and only if it is well-covered and C4-free., 11 pages, 3 figures

The stability number of a graph G, denoted by alpha(G), is the cardinality of a stable set of maximum size in G. If alpha(G-e) > alpha(G), then e is an alpha-critical edge, and if mu(G-e) < mu(G), then e is a mu-critical edge, where mu(G) is the cardinality of a maximum matching in G. G is a Koenig-Egervary graph if alpha(G) + mu(G) equals its order. Beineke, Harary and Plummer have shown that the set of alpha-critical edges of a bipartite graph is a matching. In this paper we generalize this statement to Koenig-Egervary graphs. We also prove that in a Koenig-Egervary graph alpha-critical edges are also mu-critical, and that they coincide in bipartite graphs. We obtain that for any tree its stability number equals the sum of the cardinality of the set of its alpha-critical vertices and the size of the set of its alpha-critical edges. Eventually, we characterize the Koenig-Egervary graphs enjoying this property., 15 pages, 10 figures

A square (0,1)-matrix X of order n > 0 is called fully indecomposable if there exists no integer k with 0 < k < n, such that X has a k by n-k zero submatrix. A stable set of a graph G is a subset of pairwise nonadjacent vertices. The stability number of G, denoted by $��(G)$, is the cardinality of a maximum stable set in G. A graph is called $��$-stable if its stability number remains the same upon both the deletion and the addition of any edge. We show that a connected bipartite graph has exactly two maximum stable sets that partition its vertex set if and only if its reduced adjacency matrix is fully indecomposable. We also describe a decomposition structure of $��$-stable bipartite graphs in terms of their reduced adjacency matrices. On the base of these findings we obtain both new proofs for a number of well-known theorems on the structure of matrices due to Brualdi, Marcus and Minc, Dulmage and Mendelsohn, and some generalizations of these statements. Several new results on $��$-stable bipartite graphs and their corresponding reduced adjacency matrices are presented, as well. Two kinds of matrix product are also considered (namely, Boolean product and Kronecker product), and their corresponding graph operations. As a consequence, we obtain a strengthening of one Lewin's theorem claiming that the product of two fully indecomposable matrices is a fully indecomposable matrix., A preliminary version of this paper has been presented at the Tenth Haifa Matrix Theory Conference, January 1998, Technion, Haifa, Israel; 16 pages, 5 figures

The stability number of a graph G, denoted by alpha(G), is the cardinality of a maximum stable set, and mu(G) is the cardinality of a maximum matching in G. If alpha(G) + mu(G) equals its order, then G is a Koenig-Egervary graph. We call G an $��$-square-stable graph, shortly square-stable, if alpha(G) = alpha(G*G), where G*G denotes the second power of G. These graphs were first investigated by Randerath and Wolkmann. In this paper we obtain several new characterizations of square-stable graphs. We also show that G is an square-stable Koenig-Egervary graph if and only if it has a perfect matching consisting of pendant edges. Moreover, we find that well-covered trees are exactly square-stable trees. To verify this result we give a new proof of one Ravindra's theorem describing well-covered trees., 13 pages, 6 figures

The stability number of a graph G, is the cardinality of a stable set of maximum size in G. If the stability number of G remains the same upon the addition of any edge, then G is called $��^{+}$-stable. G is a K��nig-Egervary graph if its order equals the sum of its stability number and the cardinality of a maximum matching. In this paper we characterize $��^{+}$-stable K��nig-Egervary graphs, generalizing some previously known results on bipartite graphs and trees. Namely, we prove that a K��nig-Egervary graph is $��^{+}$-stable if and only if either the set of vertices belonging to no maximum stable set is empty, or the cardinality of this set equals one, and G has a perfect matching. Using this characterization we obtain several new findings on general K��nig-Egervary graphs, for example, the equality between the cardinalities of the set of vertices belonging to all maximum stable sets and the set of vertices belonging to no maximum stable set of G is a necessary and sufficient condition for a K��nig-Egervary graph G to have a perfect matching., A preliminary version of this paper has been presented at 9th SIAM Conference on Discrete Mathematics, July 1998, the University of Toronto, Toronto, Canada; 13 pages, 5 figures

The stability number of a graph G is the cardinality of a stability system of G (that is of a stable set of maximum size of G). A graph is alpha-stable if its stability number remains the same upon both the deletion and the addition of any edge. Trying to generalize some stable trees properties, we show that there does not exist any alpha-stable chordal graph, and we prove that: if G is a connected bipartite graph, then the following assertions are equivalent: G is alpha-stable; G can be written as a vertex disjoint union of connected bipartite graphs, each of them having exactly two stability systems covering its vertex set; G has perfect matchings and no edge belongs to all its perfect matchings; from each vertex of G are issuing at least two edges contained in some perfect matchings of G; any vertex of G lies on a cycle, whose edges are alternately in and not in some perfect matching; no vertex belongs to all stability systems of G, and no edge belongs to all its perfect matchings., A preliminary version of this paper has been presented at The Third Krakow Conference On Graph Theory, September 15-19, 1997, Krakow University, Kazimierz Dolny, Poland; 16 pages

Let $G$ be a finite group and $\pi$ be a permutation from $S_{n}$. We investigate the distribution of the probabilities of the equality \[ a_{1}a_{2}\cdots a_{n-1}a_{n}=a_{\pi_{1}}a_{\pi_{2}}\cdots a_{\pi_{n-1}}a_{\pi_{n}} \] when $\pi$ varies over all the permutations in $S_{n}$. The probability \[ Pr_{\pi}(G)=Pr(a_{1}a_{2}\cdots a_{n-1}a_{n}=a_{\pi_{1}}a_{\pi_{2}}\cdots a_{\pi_{n-1}}a_{\pi_{n}}) \] is identical to $Pr_{1}^{\omega}(G)$, with \[ \omega=a_{1}a_{2}...a_{n-1}a_{n}a_{\pi_{1}}^{-1}a_{\pi_{2}}^{-1}\cdots a_{\pi_{n-1}}^{-1}a_{\pi_{n}}^{-1}, \] as it is defined in \cite{DasNath1} and \cite{NathDash1}. The notion of commutativity degree, or the probability of a permutation equality $a_{1}a_{2}=a_{2}a_{1}$, for which $n=2$ and $\pi=\langle2\;\;1\rangle$, was introduced and assessed by P. Erd\"{o}s and P. Turan in \cite{ET} in 1968 and by W. H. Gustafson in \cite{G} in 1973. In \cite{G} Gustafson establishes a relation between the probability of $a_{1},a_{2}\in G$ commuting and the number of conjugacy classes in $G$. In this work we define several other parameters, which depend only on a certain interplay between the conjugacy classes of $G$, and compute the probabilities of general permutation equalities in terms of these parameters. It turns out that this probability, for a permutation $\pi$, depends only on the number $c(Gr(\pi))$ of the alternating cycles in the cycle graph $Gr(\pi)$ of $\pi$. The cycle graph of a permutation was introduced by V. Bafna and P. A. Pevzner in \cite{BP}. We describe the spectrum of the probabilities of permutation equalities in a finite group as $\pi$ varies over all the elements of $S_{n}$. This spectrum turns-out to be closely related to the partition of $n!$ into a sum of the corresponding Hultman numbers., Comment: 36 pages

The notion of tree projection provides a natural generalization for various structural decomposition methods, which have been proposed in the literature in order to single out classes of nearly-acyclic (hyper)graphs. In this paper, the mathematical properties of the notion of tree projection are overviewed, and the complexity of the basic tree projection problem (of deciding the existence of a tree projection) is depicted. In more details, a game-theoretic characterization (in terms of the Robber and Captain game [15]) for tree projections is described, which also serves to provide simple arguments for the membership in NP of the tree projection problem. Eventually, the main ideas proposed in [14] and underlying the proof of NP-hardness of the tree projection problem are discussed.