arXiv subjects

Vadim E. Levit

Publications and source records attributed to Vadim E. Levit.

At least 73 records · Page 4Linked to original sources

arXiv2011

Critical Sets in Bipartite Graphs

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 set X is critical if d(X)=d_{c}(G). For a graph G we define ker(G) as the intersection of all critical independent sets, while diadem(G) is the union of all critical independent sets. For a bipartite graph G=(A,B,E), with bipartition {A,B}, Ore defined delta(X)=d(X) for every subset X of A, while delta_0(A)=max{delta(X):X is a subset of A}. Similarly is defined delta_0(B). In this paper we prove that for every bipartite graph G=(A,B,E) the following assertions hold: d_{c}(G)=delta_0(A)+delta_0(B); ker(G)=core(G); |ker(G)|+|diadem(G)|=2*alpha(G).

cs.DM↗

arXiv2011

On the Structure of the Minimum Critical Independent Set of a Graph

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 sets. Recently, it was established that ker(G) is a subset of core(G) is true for every graph, while the corresponding equality holds for bipartite graphs. In this paper we present various structural properties of ker(G). The main finding claims that ker(G) is equal to the union of all inclusion minimal independent sets with positive difference.

cs.DM↗

arXiv2011

Vertices Belonging to All Critical Independent Sets of a Graph

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|-|N(A)|=id_{c}(G). We define core(G) as the intersection of all maximum independent sets, and ker(G)as the intersection of all critical independent sets. In this paper we prove that if a graph G is non-quasi-regularizable (i.e., there exists some independent set A, such that |A|>|N(A)|), then: ker(G) is a subset of core(G), and |ker(G)|> id_{c}(G) >= alpha(G)-mu(G) > 0.

cs.DM↗

arXiv2011

An Algorithm Computing the Core of a Konig-Egervary Graph

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. To decide whether core(G) is empty is known to be NP-hard. In this paper, we present some polynomial time algorithms finding core(G) of a Konig-Egervary graph G.

cs.DM↗

arXiv2011

Local Maximum Stable Sets Greedoids Stemmed from Very Well-Covered Graphs

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 coinsides with (V,F). It has been shown that the family local maximum stable sets of a forest T forms a greedoid on its vertex set. In this paper we demonstrate that if G is a very well-covered graph, then its family of local maximum stable sets is a greedoid if and only if G has a unique perfect matching.

cs.DM↗

arXiv2010

Weighted Well-Covered Graphs without Cycles of Length 4, 5, 6 and 7

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.

cs.DM↗

arXiv2010

Very Well-Covered Graphs of Girth at least Four and Local Maximum Stable Set Greedoids

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 $S\inΨ(G)$, 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 $S\inΨ(G)$ is a subset of a maximum stable set of $G$. In (Levit & Mandrescu, 2002) we have shown that the family $Ψ(T)$ of a forest $T$ forms a greedoid on its vertex set. The cases where $G$ is bipartite, triangle-free, well-covered, while $Ψ(G)$ is a greedoid, were analyzed in (Levit & Mandrescu, 2002),(Levit & Mandrescu, 2004),(Levit & Mandrescu, 2007), respectively. In this paper we demonstrate that if $G$ is a very well-covered graph of girth $\geq4$, then the family $Ψ(G)$ is a greedoid if and only if $G$ has a unique perfect matching.

cs.DM↗

arXiv2010

On the independence polynomial of an antiregular graph

A graph with at most two vertices of the same degree is called antiregular (Merris 2003), maximally nonregular (Zykov 1990) or quasiperfect (Behzad, Chartrand 1967). If s_{k} is the number of independent sets of cardinality k in a graph G, then I(G;x) = s_{0} + s_{1}x + ... + s_{alpha}x^{alpha} is the independence polynomial of G (Gutman, Harary 1983), where alpha = alpha(G) is the size of a maximum independent set. In this paper we derive closed formulae for the independence polynomials of antiregular graphs. In particular, we deduce that every antiregular graph A is uniquely defined by its independence polynomial I(A;x), within the family of threshold graphs. Moreover, I(A;x) is logconcave with at most two real roots, and I(A;-1) belongs to {-1,0}.

cs.DM↗

arXiv2010

When G^2 is a Konig-Egervary 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, whenever the distance between them in G is at most 2. G is a square-stable graph if it enjoys the property alpha(G)=alpha(G^2), where alpha(G) is the size of a maximum stable set in G. In this paper we show that G^2 is a Konig-Egervary graph if and only if G is a square-stable Konig-Egervary graph.

cs.DM↗

arXiv2009

A characterization of Konig-Egervary graphs using a common property of all maximum matchings

The independence number of a graph G, denoted by alpha(G), is the cardinality of an independent set of maximum size in G, while mu(G) is the size of a maximum matching in G, i.e., its matching number. G is a Konig-Egervary graph if its order equals alpha(G)+mu(G). In this paper we give a new characterization of Konig-Egervary graphs. We also deduce some properties of vertices belonging to all maximum independent sets of a Konig-Egervary graph.

cs.DM↗

arXiv2009

On Relating Edges in Graphs without Cycles of Length 4

An edge xy is relating in the graph G if there is an independent set S, containing neither x nor y, such that S_{x} and S_{y} are both maximal independent sets in G. It is an NP-complete problem to decide whether an edge is relating (Brown, Nowakowski, Zverovich, 2007). We show that the problem remains NP-complete even for graphs without cycles of length 4 and 5. On the other hand, for graphs without cycles of length 4 and 6, the problem can be solved in polynomial time.

cs.DM↗

arXiv2009

On Konig-Egervary Square-Stable Graphs

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.

math.CO↗

arXiv2009

Critical independent sets and Konig--Egervary graphs

Let alpha(G) be the cardinality of a independence set of maximum size in the graph G, while mu(G) is the size of a maximum matching. G is a Konig--Egervary graph if its order equals alpha(G) + mu(G). The set core(G) is the intersection of all maximum independent sets of G (Levit & Mandrescu, 2002). The number def(G)=|V(G)|-2*mu(G) is the deficiency of G (Lovasz & Plummer, 1986). The number d(G)=max{|S|-|N(S)|:S in Ind(G)} is the critical difference of G. An independent set A is critical if |A|-|N(A)|=d(G), where N(S) is the neighborhood of S (Zhang, 1990). In 2009, Larson showed that G is Konig--Egervary graph if and only if there exists a maximum independent set that is critical as well. In this paper we prove that: (i) d(G)=|core(G)|-|N(core(G))|=alpha(G)-mu(G)=def(G) for every Konig--Egervary graph G; (ii) G is Konig--Egervary graph if and only if every maximum independent set of G is critical.

math.CO↗

arXiv2009

A Simple Proof of an Inequality Connecting the Alternating Number of Independent Sets and the Decycling Number

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 provide an elementary proof of the inequality claiming that the absolute value of I(G;-1) is not greater than 2^phi(G), for every graph G, where phi(G) is its decycling number.

math.CO↗

arXiv2009

Greedoids on Vertex Sets of Unicycle Graphs

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.

math.CO↗

arXiv2009

The independence polynomial of a graph at -1

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.

math.CO↗

arXiv2008

Interval greedoids and families of local maximum stable sets of graphs

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.

math.CO↗

arXiv2008

Graph Operations that are Good for Greedoids

S is a local maximum stable set of a graph G, if the set S is a maximum stable set of the subgraph induced by its closed neighborhood. In (Levit, Mandrescu, 2002) we have proved that the family of all local maximum stable sets is a greedoid for every forest. The cases of bipartite graphs and triangle-free graphs were analyzed in (Levit, Mandrescu, 2004) and (Levit, Mandrescu, 2007), respectively. In this paper we give necessary and sufficient conditions for the family of all local maximum stable sets of a graph G to form a greedoid, where G is: (a) the disjoint union of a family of graphs; (b) the Zykov sum of a family of graphs, or (c) the corona X*{H_1,H_2,...,H_n} obtained by joining each vertex k of a graph X to all the vertices of a graph H_k.

math.CO↗