Vadim Levit
Ariel University, Mathematics, Faculty Member
- Clustering Algorithms, Graph Theory, Algebraic Combinatorics, Approximation Algorithms, Formal Concept Analysis (Data Mining), Games (Computer Science), and 19 moreAlgorithms, Computational Complexity, Algorithm Analysis, Experimental Mathematics, Recommender Systems, Combinatorics, Operations research and Optimization, Computer Science, Mathematics and Computer Science, Graphs Theory, Algebraic Graph Theory, Adi Jarden, Ariel University, Critical Independent Set, Multi Peg Tower of Hanoi, Discrete Mathematics, Group Theory, Number Theory, and Computer Science Educationedit
- I am a professor of Mathematics and Algorithmics.edit
Abstract 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... more
Abstract 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 by S ∪ N ( S ) , where N(S) is the neighborhood of S. If I is stable and there is a matching from N(I) into I, then I is a crown of order | I | + | N ( I ) | , and we write I ∈ Crown ( G ) [F. N. Abu-Khzam, M. R. Fellows, M. A. Langston, W. H. Suters, Crown structures for vertex cover kernelization, Theory of Comput. Syst. 41 (2007) 411–430]. In this paper we show that Crown ( G ) ⊆ Ψ ( G ) holds for every graph, while Crown ( G ) = Ψ ( G ) is true for bipartite and very well-covered graphs. For general graphs, it is NP-complete to decide if a graph has a crown of a given order [C. Sloper. Techniques in Parameterized Algorithm Design. Ph.D. thesis, Department of Computer Science, University of Bergen, Norway, 2005]. We prove that in a bipartite graph G with a unique perfect matching, there exist crowns of every possible even order.
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Research Interests: Electricity and Heuristic
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The stability number of a graph G, denoted by � (G), is the cardinality of a stable set of maximum size in G. If its stability number remains the same upon the addition of any edge, then G is called � + -stable. G is aK # onig–Egerva E)... more
The stability number of a graph G, denoted by � (G), is the cardinality of a stable set of maximum size in G. If its stability number remains the same upon the addition of any edge, then G is called � + -stable. G is aK # onig–Egerva E) of order at least two is � + -stable if and only if G has a perfect matching and | {V − S: S ∈ � (G)}| 6 1 (where � (G) denotes the family of all maximum stable sets of G). We also show that the equality | {V − S: S ∈ � (G)}| = | {S: S ∈ � (G)}| is a necessary and su>cient condition for a K# onig–Egerv&ary graph G to have a perfect matching. Finally, we describe the two following types of � + -stable K#onig–Egerv&ary graphs: those with | {S: S ∈ � (G)}| = 0 and | {S: S ∈ � (G)}| = 1, respectively. c
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Abstract 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... more
Abstract 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 by S ∪ N ( S ) , where N(S) is the neighborhood of S. If I is stable and there is a matching from N(I) into I, then I is a crown of order | I | + | N ( I ) | , and we write I ∈ Crown ( G ) [F. N. Abu-Khzam, M. R. Fellows, M. A. Langston, W. H. Suters, Crown structures for vertex cover kernelization, Theory of Comput. Syst. 41 (2007) 411–430]. In this paper we show that Crown ( G ) ⊆ Ψ ( G ) holds for every graph, while Crown ( G ) = Ψ ( G ) is true for bipartite and very well-covered graphs. For general graphs, it is NP-complete to decide if a graph has a crown of a given order [C. Sloper. Techniques in Parameterized Algorithm Design. Ph.D. thesis, Department of Computer Science, University of Bergen, Norway, 2005]. We prove that in a bipartite graph G with a unique perfect matching, there exist crowns of every possible even order.
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In this paper we show that for any integer $alpha$ > 7, there exists a (dis)connected well-covered graph $G$ with $alpha$ = alpha(G), whose independence polynomial is not unimodal. In addition, we present a number of sufficient conditions... more
In this paper we show that for any integer $alpha$ > 7, there exists a (dis)connected well-covered graph $G$ with $alpha$ = alpha(G), whose independence polynomial is not unimodal. In addition, we present a number of sufficient conditions for a graph $G$ with alpha(G) < 7 to have unimodal independence polynomial.
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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... more
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 reduced to the construction of the shortest expression. The advantage of this method, compared with classical numeric algorithms, is its stability and faster reaction to data renewal. These problems are solved with reference to two kinds of n-node digraphs: Fibonacci graphs and complete source-target directed acyclic graphs. \(O(n^{2})\) and \(O\left( 2^{\left\lceil \log _{2}n\right\rceil ^{2}-\left\lceil \log _{2}n\right\rceil }\right) \) complexity algorithms, respectively, are provided in these cases.
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A graph is well-covered if every maximal stable set is also maximum. One theorem of Ravindra 16] asserts that a tree of order at least two i s w ell-covered if and only if it has a perfect matching consisting only of pendant edges. The... more
A graph is well-covered if every maximal stable set is also maximum. One theorem of Ravindra 16] asserts that a tree of order at least two i s w ell-covered if and only if it has a perfect matching consisting only of pendant edges. The edge-join of two connected graphs G1 G 2 is the graph G1 G2 obtained by adding an edge joining two v ertices belonging to G1 G 2, respectively. If both newly adjacent v ertices are of degree 2 in G1 G 2, respectively, then G1 G2 is an internal edge-join of G1 G 2. In this paper we show that any w ell-covered tree, having at least three vertices, can be recursively constructed using the internal edge-join operation. We also characterize well-covered trees in terms of distances between their vertices and their pendant v ertices.
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An antimatroid is an accessible set system closed under union [2]. An algorithmic characterization of antimatroids based on the language definition was introduced in [3]. Later, another algorithmic characterization of antimatroids which... more
An antimatroid is an accessible set system closed under union [2]. An algorithmic characterization of antimatroids based on the language definition was introduced in [3]. Later, another algorithmic characterization of antimatroids which depicted them as set systems was developed in [4]. While classical examples of antimatroids connect them with posets, chordal graphs, convex geometries, etc., a game theory gives a framework, in which antimatroids are considered as permission structures for coalitions [1]. A poly-antimatroid ...
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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 $G$ \[... more
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 $s_k$ denotes the number of independent sets of cardinality $k$ in the graph $G$. A comprehensive survey on the subject is due to Levit and Mandrescu, where some recursive formulas are allowing to calculate the independence polynomial. A direct implementation of these recursions does not bring about an efficient algorithm. Yosef, Mizrachi, and Kadrawi developed an efficient way for computing the independence polynomials of trees with $n$ vertices, such that a database containing all of the independence polynomials of all the trees with up to $n-1$ vertices is required. This approach is not suitable for big trees, as an extensive database is needed...
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114 M. Korenblit and VE Levit G-''au> M F 1 1 2 3------n-3 n-2 n-1 n 1 2'3-----'n-3 13-2 {£ 1'11 F03 1 2 3 rr~ r~-n—3 n-2 n-1 11 Figure 1: Generalized Fibonacci graphs of degrees 1, 2, and 3. 2 The Structure of... more
114 M. Korenblit and VE Levit G-''au> M F 1 1 2 3------n-3 n-2 n-1 n 1 2'3-----'n-3 13-2 {£ 1'11 F03 1 2 3 rr~ r~-n—3 n-2 n-1 11 Figure 1: Generalized Fibonacci graphs of degrees 1, 2, and 3. 2 The Structure of Mincuts in an F G3 We denote an n-vertex FG3 by FG3 (n) and the set of all its mincuts by CF3 (n). The set of all mincuts of FG3 may be divided into four nonintersecting subgroups denoted by CF3 (n—2, n—1, 12), CFg (n_———2, n—1, 11), CF3 (n—2mm), and CF3 (m, n——1, Mincuts of CF3 (n—2, 71—1, n ...
Page 1. Journal of Computational Methods in Sciences and Engineering 11 (2011) 271–280 271 DOI 10.3233/JCM-2011-0396 IOS Press Mincuts in generalized Fibonacci graphs of degree 3 Mark Korenblita,∗ and Vadim E. Levitb aHolon Institute of... more
Page 1. Journal of Computational Methods in Sciences and Engineering 11 (2011) 271–280 271 DOI 10.3233/JCM-2011-0396 IOS Press Mincuts in generalized Fibonacci graphs of degree 3 Mark Korenblita,∗ and Vadim E. Levitb aHolon Institute of Technology, Holon, Israel bAriel University Center of Samaria, Ariel, Israel Abstract. We investigate the structure of mincuts in an n-vertex generalized Fibonacci graph of degree 3 and show that the ...
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A graph G = (V,E) consists of a vertex set V and an edge set E, where each edge corresponds to a pair of vertices. If the edges are ordered pairs of vertices, then we call the graph directed or digraph. A vertex in a digraph is a source... more
A graph G = (V,E) consists of a vertex set V and an edge set E, where each edge corresponds to a pair of vertices. If the edges are ordered pairs of vertices, then we call the graph directed or digraph. A vertex in a digraph is a source if no edges enter it, and a sink if no edges leave it. A two-terminal directed acyclic graph (st-dag) has only one source s and only one sink t. A simple graph is a graph without multiple edges. A series-parallel (SP) graph is an st-dag defined recursively as follows: (i) A single edge (u,v) is a series-parallel graph with source u and ...