Vadim Levit

Ariel University · Department of Mathematics

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 called a Konig-Egervary graph. The number $d\left( G\right) =\max\{\left\vert A\right\vert -\left\vert N\left( A\right) \right\vert :A\subseteq V\}$...

If α(G)+μ(G)=V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha (G)+\mu (G)=\left| V\right|$$\end{document}, then G=V,E\documentclass[12pt]{minimal} \usepackage{am...

A set S⊆V is independent in a graph G=V,E if no two vertices from S are adjacent. The independence numberα(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 König–Egerváry graph (Deming in Discrete Math 27:23–33, 1979; Sterboul in J Combin Th...

A set S⊆V(G) is independent if no two vertices from S are adjacent, and by Ind(G) we mean the set of all independent sets of G. A set A∈Ind(G) is critical (and we write A∈CritIndep(G)) if A-N(A)=max{I-N(I):I∈Ind(G)} [37], where N(I) denotes the neighborhood of I. If S∈Ind(G) and there is a matching from N(S) into S, then S is a crown [1], and we wr...

Let G be a finite group. Let \(\pi \) be a permutation from \(S_{n}\). We study the distribution of probabilities of equality\( a_{1}a_{2}\cdots a_{n-1}a_{n}=a_{\pi _{1}}^{\epsilon _{1} }a_{\pi _{2}}^{\epsilon _{2}}\cdots a_{\pi _{n-1}}^{\epsilon _{n-1}}a_{\pi _{n} }^{\epsilon _{n}},\) when \(\pi \) varies over all the permutations in \(S_{n}\), an...

This paper investigates relationship between algebraic expressions and graphs. Our intent is to simplify graph expressions and eventually find their shortest representations. We prove the monotonicity results allowing to assert that the length of a shortest expression of any subgraph of a given graph is not greater than the length of a shortest exp...

An independent set in a graph is a set of pairwise non-adjacent vertices. Let $\alpha(G)$ denote the cardinality of a maximum independent set in the graph $G = (V, E)$. Gutman and Harary defined the independence polynomial of $G$ \[ I(G;x) = \sum_{k=0}^{\alpha(G)}{s_k}x^{k}={s_0}+{s_1}x+{s_2}x^{2}+...+{s_{\alpha(G)}}x^{\alpha(G)}, \] where $s_k$ de...

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 determi...

Given a finite set E and an operator sigma:2^{E}-->2^{E}, two subsets X,Y of the ground set E are cospanning if sigma(X)=sigma(Y) (Korte, Lovasz, Schrader; 1991). We investigate cospanning relations on violator spaces. A notion of a violator space was introduced in (Gartner, Matousek, Rust, Skovrovn; 2008) as a combinatorial framework that encompas...

A graph [Formula: see text] is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function [Formula: see text] is defined on its vertices. Then [Formula: see text] is [Formula: see text]well-covered if all maximal independent sets are of the same weight. For every graph [Formula: see text], the set of...

A set $S\subseteq V(G)$ is independent (or stable) if no two vertices from $S$ are adjacent, and by $\mathrm{Ind}(G)$ we mean the set of all independent sets of $G$. A set $A\in\mathrm{Ind}(G)$ is critical (and we write $A\in CritIndep(G)$) if $\left\vert A\right\vert -\left\vert N(A)\right\vert =\max\{\left\vert I\right\vert -\left\vert N(I)\right...

The paper investigates relationship between algebraic expressions and graphs. Our intent is to simplify graph expressions and eventually find their shortest representations. We describe the decomposition method for generating expressions of complete st-dags (two-terminal directed acyclic graphs) and estimate the corresponding expression complexitie...

The paper proposes a symbolic technique for shortest-path problems. This technique is based on a presentation of a shortest-path algorithm as a symbolic expression. Literals of this expression are arc tags of a graph, and they are substituted for corresponding arc weights which appear in the algorithm. The search for the most efficient algorithm is...

As a kind of converse of the celebrated Erdős–Szekeres theorem, we present a necessary and sufficient condition for a sequence of length n to contain a longest increasing subsequence and a longest decreasing subsequence of given lengths x and y, respectively.

As a kind of converse of the celebrated Erd˝os-Szekeres theorem, we present a necessary and sufficient condition for a sequence of length n to contain a longest increasing subsequence and a longest decreasing subsequence of given lengths x and y, respectively.

Let G be a simple graph with vertex set V(G). A subset S of V(G) is independent if no two vertices from S are adjacent. The graph G is known to be a Konig-Egervary if alpha(G) + mu(G)= |V(G)|, where alpha(G) denotes the size of a maximum independent set and mu(G) is the cardinality of a maximum matching. Let Omega(G) denote the family of all maximu...

Let G be a finite group. Let pi be a permutation from S{n}. We study the distribution of probabilities of equality a{1} a{2} ...a{n-1}a{n}=a{pi{1}}^{epsilon{1}} a{pi_{2}}^{epsilon{2}}...a{pi{n-1}}^{epsilon_{n-1}} a_{pi_{n}}^{epsilon{n}}, when pi varies over all the permutations in S{n}, and epsilon{i} varies over the set {+1, -1}. By the paper "Hul...

Let G be a simple graph with vertex set . A set is independent if no two vertices from S are adjacent, and by we mean the family of all independent sets of G. The number is the difference of , and a set is critical if [34]. Let us recall the following definitions: [16], [5], [18], [12] [24]. [16], [5], [18], [12] [24]. In this paper we focus on int...

A set $S\subseteq V$ is \textit{independent} in a graph $G=\left( V,E\right) $ if no two vertices from $S$ are adjacent. The \textit{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 called a \textit{K\...

Let $\alpha(G)$ denote the cardinality of a maximum independent set, while $\mu(G)$ be the size of a maximum matching in the graph $G=\left(V,E\right) $. If $\alpha(G)+\mu(G)=\left\vert V\right\vert $, then $G$ is a K\"onig-Egerv\'ary graph. If $d_{1}\leq d_{2}\leq\cdots\leq d_{n}$ is the degree sequence of $G$, then the annihilation number $h\left...

A graph $G$ is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function $w$ is defined on its vertices. Then $G$ is $w$-well-covered if all maximal independent sets are of the same weight. For every graph $G$, the set of weight functions $w$ such that $G$ is $w$-well-covered is a vector space, deno...

A graph $G$ is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function $w$ is defined on its vertices. Then $G$ is $w$-well-covered if all maximal independent sets are of the same weight. For every graph $G$, the set of weight functions $w$ such that $G$ is $w$-well-covered is a vector space, deno...

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 wa...

The Krein-Milman theorem characterizes convex subsets in topological vector spaces. Convex geometries were invented as proper combinatorial abstractions of convexity. Further, they turned out to be closure spaces satisfying the Krein-Milman property. Violator spaces were introduced in an attempt to find a general framework for LP-problems. In this...

A graph is well-covered if all its maximal independent sets are of the same size (Plummer, 1970). A graph G belongs to class Wn if every n pairwise disjoint independent sets in G are included in n pairwise disjoint maximum independent sets (Staples, 1975). Clearly, W1 is the family of all well-covered graphs. Staples showed a number of ways to buil...

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 wa...

Berge's Lemma says that for each independent set S and maximum independent set X, there is a matching from S−X into X−S, namely, a function of S−X into X−S such that (s,f(s)) is an edge for each s∈S−X. Levit and Mandrescu prove A Set and Collection Lemma. It is a strengthening of Berge's Lemma, by which one can obtain a matching M:S−⋂Γ→⋃Γ−S, where...

The paper investigates relationship between algebraic expressions and graphs. Our intention is to simplify graph expressions and eventually find their shortest representations. We prove the decomposition lemma which asserts that the shortest expression of a subgraph of a graph G is not larger than the shortest expression of G. Using this finding, w...

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...

A graph is well-covered if all its maximal independent sets are of the same cardinality (Plummer, 1970). If G is a well-covered graph, has at least two vertices, and G-v is well-covered for every vertex v, then G is a 1-well-covered graph (Staples, 1975). We call G a {\lambda}-quasi-regularizable graph if {\lambda} |S| =< |N(S)| for every independe...

The paper investigates relationships between algebraic expressions and graphs. Using the decomposition method we generate special simultaneous systems of linear recurrences for sizes of graph expressions. We propose techniques which provide closed-form solutions for these systems.

A graph is well-covered if all its maximal independent sets are of the same size (Plummer, 1970). A well-covered graph is 1-well-covered if the deletion of every vertex leaves a graph which is well-covered as well (Staples, 1975). A graph G belongs to class Wn if every n pairwise disjoint independent sets in G are included in n pairwise disjoint ma...

A graph G is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function w is defined on its vertices. Then G is w-well-covered if all maximal independent sets are of the same weight. For every graph G, the set of weight functions w such that G is w-well-covered is a vector space, denoted WCW(G). Let...

Violator Spaces were introduced by J. Matousek et al. in 2008 as generalization of Linear Programming problems. Convex geometries were invented by Edelman and Jamison in 1985 as proper combinatorial abstractions of convexity. Convex geometries are defined by anti-exchange closure operators. We investigate an interrelations between violator spaces a...

A graph is well-covered if all its maximal independent sets are of the same size (Plummer, 1970). A well-covered graph is 1-well-covered if the deletion of any one vertex leaves a graph, which is well-covered as well (Staples, 1975). A graph G belongs to class Wn if every n pairwise disjoint independent sets in G are included in n pairwise disjoint...

A set S⊆V(G) is stable (or independent) if no two vertices from S are adjacent. Let Ψ(G) be the family of all local maximum stable sets [V. E. Levit, E. Mandrescu, A new greedoid: the family of local maximum stable sets of a forest, Discr. Appl. Math. 124 (2002) 91–101] of graph G, i.e., S∈Ψ(G) if S is a maximum stable set of the subgraph induced b...

Violator Spaces were introduced by J. Matou\v{s}ek et al. in 2008 as generalization of Linear Programming problems. Convex geometries were invented by Edelman and Jamison in 1985 as proper combinatorial abstractions of convexity. Convex geometries are defined by anti-exchange closure operators. We investigate an interrelations between violator spac...

Let G be a simple graph with vertex set V(G). A set S⊆V(G) is independent if no two vertices from S are adjacent. The graph G is known to be König–Egerváry if α(G)+μ(G)=|V(G)|, where α(G) denotes the size of a maximum independent set and μ(G) is the cardinality of a maximum matching. A nonempty collection Γ of maximum independent sets is König–Eger...

Let G be a simple graph with vertex set V (G), and let Ind(G) denote the family of all independent sets of G. The number d (X) = |X| - |N(X)| is the difference of X ⊆ V (G), and a set A ∈ Ind(G) is critical whenever d(A) = max{d (I): I ∈ Ind(G)} [10]. In this paper we establish various relations between intersections and unions of all critical inde...

Let G be a simple graph with vertex set V(G). A set S is independent if no two vertices from S are adjacent. The graph G is known to be a Konig-Egervary if alpha(G)+mu(G)= |V(G)|, where alpha(G) denotes the size of a maximum independent set and mu(G) is the cardinality of a maximum matching. The number d(X)= |X|-|N(X)| is the difference of X, and a...

Let α(G) be the cardinality of a largest independent set in graph G. If sk is the number of independent sets of size k in G, then I(G;x)=s0+s1x+⋯+sαxα, α=α(G), is the independence polynomial of G (Gutman and Harary, 1983). I(G;x) is palindromic if sα-i=si for each i∈(0,1,...,⌊α/2⌋). The corona of G and H is the graph G(ring operator)H obtained by j...

The paper returns back to some known relationships between algebraic expressions and graphs. We consider digraphs called Fibonacci graphs, which give generic examples of non-series-parallel graphs. Our intent is to simplify the expressions of Fibonacci graphs and eventually find their shortest representations. With this end in view, we survey the n...

Let G be a simple graph with vertex set V(G). A subset S of V(G) is independent if no two vertices from S are adjacent. By Ind(G) we mean the family of all independent sets of G while core(G) and corona(G) denote the intersection and the union of all maximum independent sets, respectively. The number d(X)= |X|-|N(X)| is the difference of the set of...

Let (Formula presented.) denote the maximum size of an independent set of vertices and (Formula presented.) be the cardinality of a maximum matching in a graph (Formula presented.). A matching saturating all the vertices is a perfect matching. If (Formula presented.), then (Formula presented.) is called a König–Egerváry graph. A graph is unicyclic...

A graph G is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function w is defined on its vertices. Then G is w-well-covered if all maximal independent sets are of the same weight. For every graph G, the set of weight functions w such that G is w-well-covered is a vector space. Given an input claw-...

Let $G$ be a graph. A set $S$ of vertices in $G$ dominates the graph if every vertex of $G$ is either in $S$ or a neighbor of a vertex in $S$. Finding a minimal cardinality set which dominates the graph is an NP-complete problem. The graph $G$ is well-dominated if all its minimal dominating sets are of the same cardinality. The complexity status of...

Let $G$ be a simple graph with vertex set $V\left( G\right) $. A set $S\subseteq V\left( G\right) $ is independent if no two vertices from $S$ are adjacent, and by $\mathrm{Ind}(G)$ we mean the family of all independent sets of $G$. The number $d\left( X\right) =$ $\left\vert X\right\vert -\left\vert N(X)\right\vert $ is the difference of $X\subset...

A group valued function on a graph is called balanced if the product of its values along any cycle is equal to the identity element of the group. We compute the number of balanced functions from edges and vertices of a directed graph to a finite group considering two cases: when we are allowed to walk against the direction of an edge and when we ar...

We discuss functions from edges and vertices of an undirected graph to an Abelian group. Such functions, when the sum of their values along any cycle is zero, are called balanced labelings. The set of balanced labelings forms an Abelian group. We study the structure of this group and the structure of two other groups, closely related to it: the sub...

Let $G$ be a finite group and $\pi$ be a permutation from $S_n$. We investigate and compute the probability of the equality $a_1a_2 \cdots a_{n-1}a_n=a_{\pi_1}a_{\pi_2} \cdots a_{\pi_{n-1}}a_{\pi_{n}}$ in $G$. The probability of a permutation equality $a_1a_2=a_2a_1$, for which $n=2$ and $\pi=\langle2\;\;1\rangle$, was computed by W. H. Gustafson i...

We verify the conjectures of Mahadev–Peled–Sun and of Orlin, both related to equistable graphs, for the classes of simplicial, very well-covered and line graphs. Our results are based on the combinatorial features of triangle graphs and general partition graphs. In particular, we obtain several equivalent characterizations of equistable simplicial...

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 co...

A graph G is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function w is defined on its vertices. Then G is w-well-covered if all maximal independent sets are of the same weight. For every graph G, the set of weight functions w such that G is w-well-covered is a vector space, denoted WCW(G). Let...

The notion of "antimatroid with repetition" was conceived by Bjorner, Lovasz and Shor in 1991 as an extension of the notion of antimatroid in the framework of non-simple languages. Further they were investigated by the name of "poly-antimatroids" (Nakamura, 2005, Kempner & Levit, 2007), where the set system approach was used. If the underlying set...

Let psi (G) be the family of all local maximum stable sets of graph G, i.e., S epsilon psi(G) if S is a maximum stable set of the subgraph induced by S boolean OR N(S), where N(S) is the neighborhood of S. It was shown that psi(G) is a greedoid for every forest G [15]. The cases of bipartite graphs, triangle-free graphs, and well-covered graphs, we...

A graph is called unicyclic if it owns only one cycle. A matching M is called uniquely restricted in a graph G if it is the unique perfect matching of the subgraph induced by the vertices that M saturates. Clearly, μr (G) ≤ μ(G), where μr (G) denotes the size of a maximum uniquely restricted matching, while μ(G) equals the matching number of G. In...

The independence number of a graph G, denoted by α(G), is the cardinality of a maximum independent set, and μ(G) is the size of a maximum matching in G. If α(G) + μ(G) equals its order, then G is a König–Egerváry graph. The square of a graph G is the graph G 2 with the same vertex set as in G, and an edge of G 2 is joining two distinct vertices, wh...

For a graph GG let α(G),μ(G)α(G),μ(G), and τ(G)τ(G) denote its independence number, matching number, and vertex cover number, respectively. If α(G)+μ(G)=|V(G)|α(G)+μ(G)=|V(G)| or, equivalently, μ(G)=τ(G)μ(G)=τ(G), then GG is a König–Egerváry graph.In this paper we give a new characterization of König–Egerváry graphs.

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.

The paper is devoted to the methods of solving simultaneous recurrences. Specifically, we discuss transformation of matrix recurrences to regular recurrences and propose a way of solving special matrix recurrences of order three by their decomposition to matrix recurrences of order two.

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 g...

The paper investigates relationship between algebraic expressions and graphs. We consider a digraph called a Fibonacci graph which gives a generic example of non-series-parallel graphs. Our intention in this paper is to simplify the expressions of Fibonacci graphs and eventually find their shortest representations. With that end in view, we describ...

An antimatroid is an accessible set system closed under union. A poset antimatroid is a particular case of antimatroid, which is formed by the lower sets of a poset. Feasible sets in a poset antimatroid ordered by inclusion form a distributive lattice, and every distributive lattice can be formed in this way. We introduce the polydimension of an an...

An independent set in a graph is a set of pairwise non-adjacent vertices, and a(G) is the size of a maximum independent set in the graph G. If s_{k} is the number of independent sets of cardinality k in G, then I(G;x)=s_0+s_1*x+s_2*x^2+...+s_a*x^a,a=a(G), is called the independence polynomial of G (I. Gutman and F. Harary, 1983). If s_{a-i}=f(i)*s_...

If by s k is denoted the number of independent sets of cardinality k in a graph G, then \({I(G;x)=s_{0}+s_{1}x+\cdots+s_{\alpha}x^{\alpha}}\) is the independence polynomial of G (Gutman and Harary in Utilitas Mathematica 24:97–106, 1983), where α = α(G) is the size of a maximum independent set. The inequality |I (G; −1)| ≤ 2ν(G), where ν(G) is the...

We discuss functions from the edges and vertices of a directed graph to an Abelian group. Such functions, when the sum of their values along any cycle is zero, are called balanced and form an Abelian group. We study this group in two cases: when we allowed to walk against the direction of an edge taking the opposite value of the function and when w...

We discuss functions from edges and vertices of an undirected graph to an Abelian group. Such functions, when the sum of their values along any cycle is zero, are called balanced labelings. The set of balanced labelings forms an Abelian group. We study the structure of this group and the structure of two closely related to it groups: the subgroup o...

Let G = (V, E) be an undirected graph with possible multiple edges and loops (a multigraph). Let A be an Abelian group. In this work we study the following topics: 1) A function f:E → A is called balanced if the sum of its values along every closed truncated trail of G is zero. By a truncated trail we mean a trail without the last vertex. The set H...

A graph G is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function w is defined on its vertices. Then G is w-well-covered if all maximal independent sets are of the same weight. For every graph G, the set of weight functions w such that G is w-well-covered is a vector space. Given an input graph...

A graph G = (V,E) is called equistable if there exist a positive integer t and a weight function \(w:V \longrightarrow \mathbb{N}\) such that S ⊆ V is a maximal stable set of G if and only if w(S) = t. The function w, if exists, is called an equistable function of G. No combinatorial characterization of equistable graphs is known, and the complexit...

Our work is devoted to investigation of "multi" versions of antimatroids (dual of convex geometries). These discrete structures are present in many fields of mathematical social sciences. For instance, we can quote the theory of choice, where a first attempt to connect choice functions and closure operators appears in (Malishevski, 1994). Recently,...

A maximum stable set in a graph G is a stable set of maximum cardinality. S is a local maximum stable set of G, and we write S∈Ψ(G), if S is a maximum stable set of the subgraph induced by S∪N(S), where N(S) is the neighborhood of S. A greedoid (V,ℱ) is called a local maximum stable set greedoid if there exists a graph G=(V,E) such that ℱ=Ψ(G). G....

The path spectrum of a graph is the set of lengths of all maximal paths in the graph. A set S of positive lengths is tree spectral if it is the path spectrum of a tree. We show that for each even integer s⩾2s⩾2 at least 34.57% of all subsets of the set {2,3,…,s}{2,3,…,s} are tree spectral, and for each odd integer s⩾2s⩾2 at least 27.44% of all subs...

We investigate the structure of mincuts in an n-vertex generalized Fibonacci graph of degree 3 and show that the number |CF3(n)| of mincuts in this graph is equal to |CF3 (n - 1)| + |CF3(n - 2)| + |CF3(n - 3)|-|CF3(n -4)|-|CF3 (n -5)| +1.

We consider maximum series-parallel graphs, propose their recursive description, and prove that every maximal series-parallel graph is maximum as well.

A set S is independent in a graph G 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)=|V|, then G=(V,E) is called a Konig-Egervary graph. The number d_{c}(G)=max{|A|-|N(A)|} is called the critical difference...

An independent set in a graph is a set of pairwise non-adjacent vertices, and alpha(G) is the size of a maximum independent set in the graph G. A matching is a set of non-incident edges, while mu(G) is the cardinality of a maximum matching. If s_{k} is the number of independent sets of cardinality k in G, then I(G;x)=s_{0}+s_{1}x+s_{2}x^{2}+...+s_{...

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 ww be a linear set function defined on the vertices of GG. Then GG is ww-well-covered if all maximal independent sets of GG are of the same weight. The set of weight functions ww for wh...

A set S is independent in a graph G if no two vertices from S are adjacent. By core(G) we mean the intersection of all maximum independent sets. 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. A connected graph having only one cycle, say C, is a unicyclic graph. I...

Let G=(V,E). A set S is independent if no two vertices from S are adjacent. The number d(X)= |X|-|N(X)| is the difference of X, and an independent set A is critical if d(A) = max{d(I):I is an independent set}. Let us recall that ker(G) is the intersection of all critical independent sets, and core(G) is the intersection of all maximum independent s...

Let G=(V,E) be a graph. A set S is independent if no two vertices from S are adjacent, alpha(G) is the size of a maximum independent set, and core(G) is the intersection of all maximum independent sets. The number d(X)=|X|-|N(X)| is the difference of the set X, and d_{c}(G)=max{d(I):I is an independent set} is called the critical difference of G. A...

A set S of vertices is independent in a graph G if no two vertices from S are adjacent, and alpha(G) is the cardinality of a maximum independent set of G. G is called a Konig-Egervary graph if its order equals alpha(G)+mu(G), where mu(G) denotes the size of a maximum matching. By core(G) we mean the intersection of all maximum independent sets of G...

A maximum stable set in a graph G is a stable set of maximum cardinality. S is called a local maximum stable set of G if S is a maximum stable set of the subgraph induced by the closed neighborhood of S. A greedoid (V,F) is called a local maximum stable set greedoid if there exists a graph G=(V,E) such that its family of local maximum stable sets c...

Let G=(V,E) be a graph. A set S is independent if no two vertices from S are adjacent. The independence number alpha(G) is the cardinality of a maximum independent set, and mu(G) is the size of a maximum matching. The number id_{c}(G)=max{|I|-|N(I)|:I is an independent set} is called the critical independence difference of G, and A is critical if |...

A set S⊆V(G) is independent if no two vertices from S are adjacent. Let αG stand for the cardinality of a largest independent set. In this paper we prove that if Λ is a nonempty collection of maximum independent sets of a graph G, and S is an independent set, then there is a matching from S-⋂Λ into ⋃Λ-S, and S+α(G)≤⋂Λ∩S+⋃Λ∪S.Based on these findings...

An Improved Full Decomposition Algorithm for Generating Algebraic Expressions of Square Rhomboids

A \textit{maximum stable set} in a graph $G$ is a stable set of maximum cardinality. $S$ is a \textit{local maximum stable set} of $G$, and we write $S\in\Psi(G)$, if $S$ is a maximum stable set of the subgraph induced by $S\cup N(S)$, where $N(S)$ is the neighborhood of $S$. Nemhauser and Trotter Jr. (1975), proved that any $S\in\Psi(G)$ is a subs...