An independent set in a graph is a set of pairwise non-adjacent vertices. Let $\alpha(G)$ denote ... 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 computation of the independence polynomial. A direct implementation of these recursions does not bring about an efficient algorithm. In 2021, 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 databas...
Abstract A set S ⊆ V ( G ) is stable (or independent) if no two vertices from S are adjacent. Let... 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.
In many developing countries, the total electricity demand is larger than the limited generation ... more In many developing countries, the total electricity demand is larger than the limited generation capacity of power stations. Many countries adopt the common practice of routine load shedding-disconnecting entire regions from the power supply-to maintain a balance between demand and supply. Load shedding results in inflicting hardship and discomfort on households, which is even worse and hence unfair to those whose need for electricity is higher than that of others during load shedding hours. Recently, Oluwasuji et al. [2020] presented this problem and suggested several heuristic solutions. In this work, we study the electricity distribution problem as a problem of fair division, model it using the related literature on cake-cutting problems, and discuss some insights on which parts of the time intervals are allocated to each household. We consider four cases: identical demand, uniform utilities; identical demand, additive utilities; different demand, uniform utilities; different demand, additive utilities. We provide the solution for the first two cases and discuss the novel concept of q-times bin packing in relation to the remaining cases. We also show how the fourth case is related to the consensus k-division problem. One can study objectives and constraints using utilitarian and egalitarian social welfare metrics, as well as trying to keep the number of cuts as small as possible. A secondary objective can be to minimize the maximum utility-difference between agents.
An independent set in a graph is a set of pairwise nonadjacent vertices. Let α G ðÞ denote the ca... more An independent set in a graph is a set of pairwise nonadjacent vertices. Let α G ðÞ denote the cardinality of a maximum independent set in the graph G ¼ V, E ðÞ .In 1983, Gutman and Harary defined the independence polynomial of GI G; x ðÞ ¼ P α G ðÞ k¼0 s k x k ¼ s 0 þ s 1 x þ s 2 x 2 þ … þ s α G ðÞ x α G ðÞ , 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 computation of the independence polynomial. A direct implementation of these recursions does not bring about an efficient algorithm. In 2021, 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 À 1 vertices is required. This approach is not suitable for big trees, as an extensive database is needed. On the other hand, using dynamic programming, it is possible to develop an efficient algorithm that prevents repeated calculations. In summary, our dynamic programming algorithm runs over a tree in linear time and does not depend on a database.
An independent set in a graph is a collection of vertices that are not adjacent to each other. Th... more 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 α(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) = α(G) k=0 s k x k = s0 + s1x + s2x 2 + ... + s α(G) x α(G) , where s k represents the number of independent sets in G of size k. The conjecture made by Alavi, Malde, Schwenk, and Erdös 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.
The stability number α(G) of the graph G is the size of a maximum stable set of G. If s k denotes... more The stability number α(G) of the graph G is the size of a maximum stable set of G. If s k denotes the number of stable sets of cardinality k in graph G, then I(G; x) = α(G) k=0 s k x k is the independence polynomial of G (I. Gutman and F. Harary, 1983). In 1990, Y. O. Hamidoune proved that for any claw-free graph G (a graph having no induced subgraph isomorphic to K1,3), I(G; x) is unimodal, i.e., there exists some k ∈ {0, 1, ..., α(G)} such that
G is a König-Egerváry graph provided α(G) + µ(G) = |V (G)|, where µ(G) is the size of a maximum m... more G is a König-Egerváry graph provided α(G) + µ(G) = |V (G)|, where µ(G) is the size of a maximum matching and α(G) is the cardinality of a maximum stable set, [3], [22]. 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, [12]. Nemhauser and Trotter Jr. proved that any S ∈ Ψ(G) is a subset of a maximum stable set of G, [20]. In this paper we demonstrate that if S ∈ Ψ(G), the subgraph H induced by S ∪ N (S) is a König-Egerváry graph, and M is a maximum matching in H, then M is a local maximum stable set in the line graph of G.
The stability number of a graph G, denoted by � (G), is the cardinality of a stable set of maximu... 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
Let G be a graph. A set S ⊆ V (G) is independent if its elements are pairwise nonadjacent. A vert... more Let G be a graph. A set S ⊆ V (G) is independent if its elements are pairwise nonadjacent. A vertex v ∈ V (G) is shedding if for every independent set S ⊆ V (G) \ N [v] there exists u ∈ N (v) such that S ∪{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 α(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 α(G). The graph G belongs to class W2 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 W2 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 W2 class is not known. Even when the input is restricted to well-covered graphs, the complexity status of recognizing graphs in W2 is not known. In this article, we investigate the connection between shedding vertices and W2 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 W2 graphs.
A graph G is well-covered if all its maximal independent sets are of the same cardinality. Assume... more 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 G without cycles of length 4, 5, and 6, we characterize polynomially the vector space of weight functions w for which G is w-well-covered. Let B be an induced complete bipartite subgraph of G on vertex sets of bipartition BX and BY. Assume that there exists an independent set S such that each of S ∪ BX and S ∪ BY is a maximal independent set of G. Then B is a generating subgraph of G, and it produces the restriction w(BX) = w(BY). It is known that for every weight function w, if G is w-well-covered, then the above restriction is satisfied. In the special case, where BX = {x} and BY = {y}, we say that xy is a relating edge. Recognizing relating edges and generating subgraphs is an NP-complete problem. However, we provide a polynomial algorithm for recognizing generating subgraphs of an input graph without cycles of length 5, 6 and 7. We also present a polynomial algorithm for recognizing relating edges in an input graph without cycles of length 5 and 6.
Electronic Notes in Discrete Mathematics, Oct 1, 2016
Abstract A set S ⊆ V ( G ) is stable (or independent) if no two vertices from S are adjacent. Let... 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.
The stability number α(G) of the graph G is the size of a maximum stable set of G. If s k denotes... more The stability number α(G) of the graph G is the size of a maximum stable set of G. If s k denotes the number of stable sets of cardinality k in graph G, then I(G; x) = α(G) k=0 s k x k is the independence polynomial of G (I. Gutman and F. Harary, 1983). In 1990, Y. O. Hamidoune proved that for any claw-free graph G (a graph having no induced subgraph isomorphic to K1,3), I(G; x) is unimodal, i.e., there exists some k ∈ {0, 1, ..., α(G)} such that
A stable set in a graph G is a set of mutually non-adjacent vertices, α(G) is the size of a maxim... more A stable set in a graph G is a set of mutually non-adjacent vertices, α(G) is the size of a maximum stable set of G, and core(G) is the intersection of all its maximum stable sets. In this paper we demonstrate that in a tree T , of order n ≥ 2, any stable set of size ≥ n/2 contains at least one pendant vertex. Hence, we deduce that any maximum stable set in a tree contains at least one pendant vertex. We give a new proof for a theorem of Hopkins and Staton [5] characterizing strong unique trees. Using this result we show that if {A, B} is the bipartition of a tree T and S is a stable set with |S| > min{|A| , |B|}, then S contains at least a pendant vertex. Our main finding is the theorem claiming that if T is a tree of order n ≥ 2 that does not own a perfect matching (i.e., 2α(T) > n), then at least two pendant vertices an even distance apart belong to core(T). While it is known that if G is a connected bipartite graph of order n ≥ 2, then |core(G)| = 1 (see Levit, Mandrescu [7]), our new statement reveals an additional structure of the intersection of all maximum stable sets of a tree. The above assertions give refining of one assertion of Hammer, Hansen and Simeone [4] stating that if a graph G is of order less than 2α(G), then core(G) is non-empty, and also of a result of Jamison [6], Gunter, Hartnel and Rall [3], and Zito [10], saying that for a tree T of order at least two, |core(T)| = 1.
The stability number α(G) of a graph G is the size of a maximum stable set of G, core(G) = ∩{S : ... more The stability number α(G) of a graph G is the size of a maximum stable set of G, core(G) = ∩{S : S is a maximum stable in G}, and ξ(G) = |core(G)|. In this paper we prove that for a graph G without isolated vertices, the following assertions are true: (i) if ξ(G) ≤ 1, then G is quasi-regularizable; (ii) if G is of order n and α(G) > (n + k − 1)/2, for some k ≥ 1, then ξ(G) ≥ k + 1, and ξ(G) ≥ k + 2, whenever n + k − 1 is even. The last finding is a strengthening of a result of Hammer, Hansen, and Simeone, which states that α(G) > n/2 implies ξ(G) ≥ 1. In the case of König-Egerváry graphs, i.e., for graphs enjoying α(G) + µ(G) = n, where µ(G) is the maximum size of a matching of G, we prove that |core(G)| > |N (core(G))| is a necessary and sufficient condition for α(G) > n/2. Moreover, for bipartite graphs without isolated vertices, ξ(G) ≥ 2 is equivalent to α(G) > n/2. We also show that Hall's marriage Theorem is valid for König-Egerváry graphs, and, it is sufficient to check Hall's condition only for one specific stable set, namely, for core(G).
In this paper we show that for any integer $alpha$ > 7, there exists a (dis)connected well-covere... 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.
An independent set in a graph is a set of pairwise non-adjacent vertices. Let $\alpha(G)$ denote ... 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 computation of the independence polynomial. A direct implementation of these recursions does not bring about an efficient algorithm. In 2021, 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 databas...
Abstract A set S ⊆ V ( G ) is stable (or independent) if no two vertices from S are adjacent. Let... 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.
In many developing countries, the total electricity demand is larger than the limited generation ... more In many developing countries, the total electricity demand is larger than the limited generation capacity of power stations. Many countries adopt the common practice of routine load shedding-disconnecting entire regions from the power supply-to maintain a balance between demand and supply. Load shedding results in inflicting hardship and discomfort on households, which is even worse and hence unfair to those whose need for electricity is higher than that of others during load shedding hours. Recently, Oluwasuji et al. [2020] presented this problem and suggested several heuristic solutions. In this work, we study the electricity distribution problem as a problem of fair division, model it using the related literature on cake-cutting problems, and discuss some insights on which parts of the time intervals are allocated to each household. We consider four cases: identical demand, uniform utilities; identical demand, additive utilities; different demand, uniform utilities; different demand, additive utilities. We provide the solution for the first two cases and discuss the novel concept of q-times bin packing in relation to the remaining cases. We also show how the fourth case is related to the consensus k-division problem. One can study objectives and constraints using utilitarian and egalitarian social welfare metrics, as well as trying to keep the number of cuts as small as possible. A secondary objective can be to minimize the maximum utility-difference between agents.
An independent set in a graph is a set of pairwise nonadjacent vertices. Let α G ðÞ denote the ca... more An independent set in a graph is a set of pairwise nonadjacent vertices. Let α G ðÞ denote the cardinality of a maximum independent set in the graph G ¼ V, E ðÞ .In 1983, Gutman and Harary defined the independence polynomial of GI G; x ðÞ ¼ P α G ðÞ k¼0 s k x k ¼ s 0 þ s 1 x þ s 2 x 2 þ … þ s α G ðÞ x α G ðÞ , 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 computation of the independence polynomial. A direct implementation of these recursions does not bring about an efficient algorithm. In 2021, 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 À 1 vertices is required. This approach is not suitable for big trees, as an extensive database is needed. On the other hand, using dynamic programming, it is possible to develop an efficient algorithm that prevents repeated calculations. In summary, our dynamic programming algorithm runs over a tree in linear time and does not depend on a database.
An independent set in a graph is a collection of vertices that are not adjacent to each other. Th... more 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 α(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) = α(G) k=0 s k x k = s0 + s1x + s2x 2 + ... + s α(G) x α(G) , where s k represents the number of independent sets in G of size k. The conjecture made by Alavi, Malde, Schwenk, and Erdös 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.
The stability number α(G) of the graph G is the size of a maximum stable set of G. If s k denotes... more The stability number α(G) of the graph G is the size of a maximum stable set of G. If s k denotes the number of stable sets of cardinality k in graph G, then I(G; x) = α(G) k=0 s k x k is the independence polynomial of G (I. Gutman and F. Harary, 1983). In 1990, Y. O. Hamidoune proved that for any claw-free graph G (a graph having no induced subgraph isomorphic to K1,3), I(G; x) is unimodal, i.e., there exists some k ∈ {0, 1, ..., α(G)} such that
G is a König-Egerváry graph provided α(G) + µ(G) = |V (G)|, where µ(G) is the size of a maximum m... more G is a König-Egerváry graph provided α(G) + µ(G) = |V (G)|, where µ(G) is the size of a maximum matching and α(G) is the cardinality of a maximum stable set, [3], [22]. 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, [12]. Nemhauser and Trotter Jr. proved that any S ∈ Ψ(G) is a subset of a maximum stable set of G, [20]. In this paper we demonstrate that if S ∈ Ψ(G), the subgraph H induced by S ∪ N (S) is a König-Egerváry graph, and M is a maximum matching in H, then M is a local maximum stable set in the line graph of G.
The stability number of a graph G, denoted by � (G), is the cardinality of a stable set of maximu... 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
Let G be a graph. A set S ⊆ V (G) is independent if its elements are pairwise nonadjacent. A vert... more Let G be a graph. A set S ⊆ V (G) is independent if its elements are pairwise nonadjacent. A vertex v ∈ V (G) is shedding if for every independent set S ⊆ V (G) \ N [v] there exists u ∈ N (v) such that S ∪{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 α(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 α(G). The graph G belongs to class W2 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 W2 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 W2 class is not known. Even when the input is restricted to well-covered graphs, the complexity status of recognizing graphs in W2 is not known. In this article, we investigate the connection between shedding vertices and W2 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 W2 graphs.
A graph G is well-covered if all its maximal independent sets are of the same cardinality. Assume... more 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 G without cycles of length 4, 5, and 6, we characterize polynomially the vector space of weight functions w for which G is w-well-covered. Let B be an induced complete bipartite subgraph of G on vertex sets of bipartition BX and BY. Assume that there exists an independent set S such that each of S ∪ BX and S ∪ BY is a maximal independent set of G. Then B is a generating subgraph of G, and it produces the restriction w(BX) = w(BY). It is known that for every weight function w, if G is w-well-covered, then the above restriction is satisfied. In the special case, where BX = {x} and BY = {y}, we say that xy is a relating edge. Recognizing relating edges and generating subgraphs is an NP-complete problem. However, we provide a polynomial algorithm for recognizing generating subgraphs of an input graph without cycles of length 5, 6 and 7. We also present a polynomial algorithm for recognizing relating edges in an input graph without cycles of length 5 and 6.
Electronic Notes in Discrete Mathematics, Oct 1, 2016
Abstract A set S ⊆ V ( G ) is stable (or independent) if no two vertices from S are adjacent. Let... 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.
The stability number α(G) of the graph G is the size of a maximum stable set of G. If s k denotes... more The stability number α(G) of the graph G is the size of a maximum stable set of G. If s k denotes the number of stable sets of cardinality k in graph G, then I(G; x) = α(G) k=0 s k x k is the independence polynomial of G (I. Gutman and F. Harary, 1983). In 1990, Y. O. Hamidoune proved that for any claw-free graph G (a graph having no induced subgraph isomorphic to K1,3), I(G; x) is unimodal, i.e., there exists some k ∈ {0, 1, ..., α(G)} such that
A stable set in a graph G is a set of mutually non-adjacent vertices, α(G) is the size of a maxim... more A stable set in a graph G is a set of mutually non-adjacent vertices, α(G) is the size of a maximum stable set of G, and core(G) is the intersection of all its maximum stable sets. In this paper we demonstrate that in a tree T , of order n ≥ 2, any stable set of size ≥ n/2 contains at least one pendant vertex. Hence, we deduce that any maximum stable set in a tree contains at least one pendant vertex. We give a new proof for a theorem of Hopkins and Staton [5] characterizing strong unique trees. Using this result we show that if {A, B} is the bipartition of a tree T and S is a stable set with |S| > min{|A| , |B|}, then S contains at least a pendant vertex. Our main finding is the theorem claiming that if T is a tree of order n ≥ 2 that does not own a perfect matching (i.e., 2α(T) > n), then at least two pendant vertices an even distance apart belong to core(T). While it is known that if G is a connected bipartite graph of order n ≥ 2, then |core(G)| = 1 (see Levit, Mandrescu [7]), our new statement reveals an additional structure of the intersection of all maximum stable sets of a tree. The above assertions give refining of one assertion of Hammer, Hansen and Simeone [4] stating that if a graph G is of order less than 2α(G), then core(G) is non-empty, and also of a result of Jamison [6], Gunter, Hartnel and Rall [3], and Zito [10], saying that for a tree T of order at least two, |core(T)| = 1.
The stability number α(G) of a graph G is the size of a maximum stable set of G, core(G) = ∩{S : ... more The stability number α(G) of a graph G is the size of a maximum stable set of G, core(G) = ∩{S : S is a maximum stable in G}, and ξ(G) = |core(G)|. In this paper we prove that for a graph G without isolated vertices, the following assertions are true: (i) if ξ(G) ≤ 1, then G is quasi-regularizable; (ii) if G is of order n and α(G) > (n + k − 1)/2, for some k ≥ 1, then ξ(G) ≥ k + 1, and ξ(G) ≥ k + 2, whenever n + k − 1 is even. The last finding is a strengthening of a result of Hammer, Hansen, and Simeone, which states that α(G) > n/2 implies ξ(G) ≥ 1. In the case of König-Egerváry graphs, i.e., for graphs enjoying α(G) + µ(G) = n, where µ(G) is the maximum size of a matching of G, we prove that |core(G)| > |N (core(G))| is a necessary and sufficient condition for α(G) > n/2. Moreover, for bipartite graphs without isolated vertices, ξ(G) ≥ 2 is equivalent to α(G) > n/2. We also show that Hall's marriage Theorem is valid for König-Egerváry graphs, and, it is sufficient to check Hall's condition only for one specific stable set, namely, for core(G).
In this paper we show that for any integer $alpha$ > 7, there exists a (dis)connected well-covere... 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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Papers by Vadim Levit