Vadim E. Levit

Combinatorial properties of the family of maximum stable sets of a graph

Abstract 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 set in G}, and ξ(G)=|core(G)|. In this paper we prove that for a graph G the following assertions are true: (i) if G has no isolated vertices, and ξ(G)⩽1, then G is quasi-regularizable; (ii) if the order of G is n, and α(G)>(n+k−min{1,|N(core(G))|})/2, for some k⩾1, then ξ(G)⩾k+1; moreover, if n+k−min{1,|N(core(G))|} is even, then ξ(G)⩾k+2. The last finding is a strengthening of a result of Hammer, Hansen, and Simeone, which states that ξ(G)⩾1 is true whenever α(G)>n/2. In the case of Konig–Egervary graphs, i.e., for graphs enjoying the equality 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. Furthermore, for bipartite graphs without isolated vertices, ξ(G)⩾2 is equivalent to α(G)>n/2. We also show that Halls Marriage Theorem is true for Konig–Egervary graphs, and, it is sufficient to check Halls condition only for one specific stable set, namely for core(G).

Independence polynomials of well-covered graphs: generic counterexamples for the unimodality conjecture

A graph G is well-covered if all its maximal stable sets have the same size, denoted by α(G) [M.D. Plummer, Some covering concepts in graphs, Journal of Combinatorial Theory 8 (1970) 91-98]. If sk denotes the number of stable sets of cardinality k in graph G, and α(G) is the size of a maximum stable set, then I(G; x) = Σk=0α(G) skxk is the independence polynomial of G [I. Gutman, F. Harary, Generalizations of the matching polynomial, Utilitas Mathematica 24 (1983) 97-106]. J.I. Brown, K. Dilcher and R.J. Nowakowski [Roots of independence polynomials of well-covered graphs, Journal of Algebraic Combinatorics 11 (2000) 197-210] conjectured that I(G; x) is unimodal (i.e., there is some j ∈ {0, 1,..., α(G)} such that s0 ≤ ... ≤ sj-1 ≤ sj ≥ sj+1 ≥ ... ≥ sα(G)) for any well-covered graph G. T.S. Michael and W.N. Traves [Independence sequences of well-covered graphs: non-unimodality and the roller-coaster conjecture, Graphs and Combinatorics 19 (2003) 403-411] proved that this assertion is true for α(G) ≤ 3, while for α(G) ∈ {4,5,6,7} they provided counterexamples.In this paper we show that for any integer α ≥ 8, there exists a connected well-covered graph G with α = α(G), whose independence polynomial is not unimodal. In addition, we present a number of sufficient conditions for a graph G with α(G) ≤ 6 to have the unimodal independence polynomial.

On unimodality of independence polynomials of some well-covered trees

The stability number α(G) of the graph G is the size of a maximum stable set of G. If sk denotes the number of stable sets of cardinality k in graph G, then I(G; x) = Σk=0α(G) skxk 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 s0 ≤ s1 ≤ ... ≤ sk-1 ≤ sk ≥ sk+1 ≥ ... ≥ sα(G). Y. Alavi, P.J. Malde, A.J. Schwenk, and P. Erdos (1987) asked whether for trees the independence polynomial is unimodal. J. I. Brown, K. Dilcher and R.J. Nowakowski (2000) conjectured that I(G; x) is unimodal for any well-covered graph G (a graph whose all maximal independent sets have the same size). Michael and Traves (2002) showed that this conjecture is true for well-covered graphs with α(G) ≤ 3, and provided counterexamples for α(G) ∈ {4, 5, 6, 7}. In this paper we show that the independence polynomial of any well-covered spider is unimodal and locate its mode, where a spider is a tree having at most one vertex of degree at least three. In addition, we extend some graph transformations, first introduced in [14], 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.

On α + -stable König-Egerváry graphs

The stability number of a graph G, denoted by α(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 α+-stable. G is a Konig-Egervary graph if its order equals α(G) + µ(G), where µ(G) is the size of a maximum matching in G. In this paper, we characterize α+-stable Konig-Egervary graphs, generalizing some previously known results on bipartite graphs and trees. Namely, we prove that a Konig-Egervary graph G = (V,E) of order at least two is α+-stable if and only if G has a perfect matching and |∪ {V - S: S ∈ Ω(G)}| ≤ 1 (where Ω (G) denotes the family of all maximum stable sets of G). We also show that the equality |∪{V - S: S ∈ Ω(G)} = |∪{S: S ∈ Ω(G)}| is a necessary and sufficient condition for a Konig-Egervary graph G to have a perfect matching. Finally, we describe the two following types of α+ stable Konig-Egervary graphs: those with |∪ {S: S ∈ Ω (G)}| = 0 and | ∪{S: S ∈ Ω(G)}| = 1, respectively.

Local maximum stable sets in bipartite graphs with uniquely restricted maximum matchings

A maximum stable set in a graph G is a stable set of maximum size. 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 spanned by S U N(S), where N(S) is the neighborhood of S. A matching M is uniquely restricted if its saturated vertices induce a subgraph which has a unique perfect matching, namely M itself. Nemhauser and Trotter Jr. (Math. Programming 8(1975) 232-248), proved that any S ∈ Ψ(G) is a subset of a maximum stable set of G. In Levit and Mandrescu (Discrete Appl. Math., 124 (2002) 91-101) we have shown that the family Ψ(T) of a forest T forms a greedoid on its vertex set. In this paper, we demonstrate that for a bipartite graph G, Ψ(G) is a greedoid on its vertex set if and only if all its maximum matchings are uniquely restricted.

A new Greedoid: the family of local maximum stable sets of a forest

A maximum stable set in a graph G is a stable set of maximum cardinality. S is a local maxtmum stable set if it is a maximum stable of the subgraph of G spanned by S∪N(S), where N(S) is the neighborhood of S. One theorem of Nemhauser and Trotter Jr. (Math. Programming 8 (1975) 232-248), working as a useful sufficient local optimality condition for the weighted maximum stable set problem, ensures that any local maximum stable set of G can be enlarged to a maximum stable set of G. In this paper we demonstrate that an inverse assertion is true for forests. Namely, we show that for any non-empty local maximum stable set S of a forest T there exists a local maximum stable set S1 of T, such that S1 ⊂ S and |S1|=|S|-1. Moreover, as a further strengthening of both the theorem of Nemhauser and Trotter Jr. and its inverse, we prove that the family of all local maximum stable sets of a forest forms a greedoid on its vertex set.

On algebraic expressions of series-parallel and Fibonacci graphs

The paper investigates relationship between algebraic expressions and graphs. Through out the paper we consider two kinds of digraphs: series-parallel graphs and Fibonacci graphs (which give a generic example of non-series-parallel graphs). Motivated by the fact that the most compact expressions of series-parallel graphs are read-once formulae, and, thus, of O(n) length, we propose an algorithm generating expressions of O(n2) length for Fibonacci graphs. A serious effort was made to prove that this algorithm yields expressions with a minimum number of terms. Using an interpretation of a shortest path algorithm as an algebraic expression, a symbolic approach to the shortest path problem is proposed.

Graph Theory, Computational Intelligence and Thought

Landmarks in Algorithmic Graph Theory: A Personal Retrospective.- A Higher-Order Graph Calculus for Autonomic Computing.- Algorithms on Subtree Filament Graphs.- A Note on the Recognition of Nested Graphs.- Asynchronous Congestion Games.- Combinatorial Problems for Horn Clauses.- Covering a Tree by a Forest.- Dominating Induced Matchings.- HyperConsistency Width for Constraint Satisfaction: Algorithms and Complexity Results.- Local Search Heuristics for the Multidimensional Assignment Problem.- On Distance-3 Matchings and Induced Matchings.- On Duality between Local Maximum Stable Sets of a Graph and Its Line-Graph.- On Path Partitions and Colourings in Digraphs.- On Related Edges in Well-Covered Graphs without Cycles of Length 4 and 6.- On the Cubicity of AT-Free Graphs and Circular-Arc Graphs.- O(m logn) Split Decomposition of Strongly Connected Graphs.- Path-Bicolorable Graphs.- Path Partitions, Cycle Covers and Integer Decomposition.- Properly Coloured Cycles and Paths: Results and Open Problems.- Recognition of Antimatroidal Point Sets.- Tree Projections: Game Characterization and Computational Aspects.

On α-critical edges in König-Egerváry graphs

The stability number of a graph G, denoted by @a(G), is the cardinality of a stable set of maximum size in G. If @a(G-e)>@a(G), then e is an @a-critical edge, and if @m(G-e)<@m(G), then e is a @m-critical edge, where @m(G) is the cardinality of a maximum matching in G. G is a Konig-Egervary graph if its order equals @a(G)+@m(G). Beineke, Harary and Plummer have shown that the set of @a-critical edges of a bipartite graph forms a matching. In this paper we generalize this statement to Konig-Egervary graphs. We also prove that in a Konig-Egervary graph @a-critical edges are also @m-critical, and that they coincide in bipartite graphs. For Konig-Egervary graphs, we characterize @m-critical edges that are also @a-critical. Eventually, we deduce that @a(T)=@x(T)+@h(T) holds for any tree T, and describe the Konig-Egervary graphs enjoying this property, where @x(G) is the number of @a-critical vertices and @h(G) is the number of @a-critical edges.

Unicycle Bipartite Graphs with Only Uniquely Restricted Maximum Matchings

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. G is a unicycle graph if it owns only one cycle.