Results 1  10
of
21
Evaluations of Graph Polynomials
, 2008
"... A graph polynomial p(G, ¯ X) can code numeric information about the underlying graph G in various ways: as its degree, as one of its specific coefficients or as evaluations at specific points ¯ X = ¯x0. In this paper we study the question how to prove that a given graph parameter, say ω(G), the siz ..."
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A graph polynomial p(G, ¯ X) can code numeric information about the underlying graph G in various ways: as its degree, as one of its specific coefficients or as evaluations at specific points ¯ X = ¯x0. In this paper we study the question how to prove that a given graph parameter, say ω(G), the size of the maximal clique of G, cannot be a fixed coefficient or the evaluation at any point of the Tutte polynomial, the interlace polynomial, or any graph polynomial of some infinite family of graph polynomials. Our result is very general. We give a sufficient condition in terms of the connection matrix of graph parameter f(G) which implies that it cannot be the evaluation of any graph polynomial which is invariantly definable in CMSOL, the Monadic Second Order Logic augmented with modular counting quantifiers. This criterion covers most of the graph polynomials known from the literature.
THE ENUMERATION OF VERTEX INDUCED SUBGRAPHS WITH RESPECT TO THE NUMBER OF COMPONENTS
, 2009
"... Inspired by the study of community structure in connection networks, we introduce the graph polynomial Q (G; x, y), the bivariate generating function which counts the number of connected components in induced subgraphs. We give a recursive definition of Q (G; x, y) using vertex deletion, vertex con ..."
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Inspired by the study of community structure in connection networks, we introduce the graph polynomial Q (G; x, y), the bivariate generating function which counts the number of connected components in induced subgraphs. We give a recursive definition of Q (G; x, y) using vertex deletion, vertex contraction and deletion of a vertex together with its neighborhood and prove a universality property. We relate Q (G; x, y) to other known graph invariants and graph polynomials, among them partition functions, the Tutte polynomial, the independence and matching polynomials, and the universal edge elimination polynomial introduced by I. Averbouch, B. Godlin and J.A. Makowsky (2008). We show that Q(G; x, y) is vertex reconstructible in the sense of Kelly and Ulam, discuss its use in computing residual connectedness reliability. Finally we show that the computation of Q(G; x, y) is ♯Phard, but Fixed Parameter
A most general edge elimination polynomial  thickening of edges
, 2008
"... We consider a graph polynomial ξ(G;x,y,z) introduced by Averbouch, Godlin, and Makowsky (2007). This graph polynomial simultaneously generalizes the Tutte polynomial as well as a bivariate chromatic polynomial defined by Dohmen, Pönitz and Tittmann (2003). We derive an identity which relates the gra ..."
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Cited by 7 (0 self)
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We consider a graph polynomial ξ(G;x,y,z) introduced by Averbouch, Godlin, and Makowsky (2007). This graph polynomial simultaneously generalizes the Tutte polynomial as well as a bivariate chromatic polynomial defined by Dohmen, Pönitz and Tittmann (2003). We derive an identity which relates the graph polynomial of a thicked graph (i.e. a graph with each edge replaced by k copies of it) to the graph polynomial of the original graph. As a consequence, we observe that at every point (x,y,z), except for points lying within some set of dimension 2, evaluating ξ is #Phard.
Graph polynomials: From recursive definitions . . .
, 2008
"... Many graph polynomials, such as the Tutte polynomial, the interlace polynomial and the matching polynomial, have both a recursive definition and a defining subset expansion formula. In this paper we present a general, logicbased framework which gives a precise meaning to recursive definitions of gr ..."
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Cited by 5 (4 self)
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Many graph polynomials, such as the Tutte polynomial, the interlace polynomial and the matching polynomial, have both a recursive definition and a defining subset expansion formula. In this paper we present a general, logicbased framework which gives a precise meaning to recursive definitions of graph polynomials. We then prove that in this framework every recursive definition of a graph polynomial can be converted into a subset expansion formula.
Linear Recurrence Relations for Graph Polynomials
"... on the occasion of his 85th birthday. Abstract. A sequence of graphs Gn is iteratively constructible if it can be built from an initial labeled graph by means of a repeated fixed succession of elementary operations involving addition of vertices and edges, deletion of edges, and relabelings. Let Gn ..."
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Cited by 5 (2 self)
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on the occasion of his 85th birthday. Abstract. A sequence of graphs Gn is iteratively constructible if it can be built from an initial labeled graph by means of a repeated fixed succession of elementary operations involving addition of vertices and edges, deletion of edges, and relabelings. Let Gn be a iteratively constructible sequence of graphs. In a recent paper, [27], M. Noy and A. Ribò have proven linear recurrences with polynomial coefficients for the Tutte polynomials T(Gi,x,y)=T(Gi), i.e. T(Gn+r)=p1(x,y)T(Gn+r−1)+...+ pr(x,y)T(Gn). We show that such linear recurrences hold much more generally for a wide class of graph polynomials (also of labeled or signed graphs), namely they hold for all the extended MSOLdefinable graph polynomials. These include most graph and knot polynomials studied in the literature. 1
Complexity of the BollobásRiordan Polynomial  Exceptional Points and Uniform Reductions
"... The coloured Tutte polynomial by Bollobás and Riordan is, as a generalization of the Tutte polynomial, the most general graph polynomial for coloured graphs that satisfies certain contractiondeletion identities. Jaeger, Vertigan, and Welsh showed that the classical Tutte polynomial is #Phard to ..."
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The coloured Tutte polynomial by Bollobás and Riordan is, as a generalization of the Tutte polynomial, the most general graph polynomial for coloured graphs that satisfies certain contractiondeletion identities. Jaeger, Vertigan, and Welsh showed that the classical Tutte polynomial is #Phard to evaluate almost everywhere by establishing reductions along curves and lines. We establish a similar result for the coloured Tutte polynomial on integral domains. To capture the algebraic flavour and the uniformity inherent in this type of result, we introduce a new kind of reductions, uniform algebraic reductions, that are wellsuited to investigate the evaluation complexity of graph polynomials. Our main result identifies a small, algebraic set of exceptional points and says that the evaluation problem of the coloured Tutte is equivalent for all nonexceptional points, under polynomialtime uniform algebraic reductions.
Model Theoretic Methods in Finite Combinatorics
, 2009
"... Purpose of the special session We want to bring the various aspects of the interaction between Model Theory and Finite Combinatorics to a wider audience. Altough the work on 01 laws is by now widely known, the other applications on counting functions, graph polynomials, extremal combinatorics, grap ..."
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Cited by 4 (0 self)
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Purpose of the special session We want to bring the various aspects of the interaction between Model Theory and Finite Combinatorics to a wider audience. Altough the work on 01 laws is by now widely known, the other applications on counting functions, graph polynomials, extremal combinatorics, graph minors and regularity lemmas, have not yet received their deserved attention. Background In the last twenty years several applications of Logic, in particular Model theory, to problems in Combinatorics emerged. Among them we have • 01 laws and their variations. This is well summarized in the book by J. Spencer [Spe01]. • Modular linear recurrence relations for combinatorial counting functions (The SpeckerBlatter Theorem) This theorem has remained widely unnoticed, and deserves wider attention and further study [Spe88, Fis03, FM03]
The complexity of multivariate matching polynomials
, 2007
"... We study various versions of the univariate and multivariate matching and rook polynomials. We show that there is most general multivariate matching polynomial, which is, up the some simple substitutions and multiplication with a prefactor, the original multivariate matching polynomial introduced by ..."
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We study various versions of the univariate and multivariate matching and rook polynomials. We show that there is most general multivariate matching polynomial, which is, up the some simple substitutions and multiplication with a prefactor, the original multivariate matching polynomial introduced by C. Heilmann and E. Lieb. We follow here a line of investigation which was very successfully pursued over the years by, among others, W. Tutte, B. Bollobas and O. Riordan, and A. Sokal in studying the chromatic and the Tutte polynomial. We show here that evaluating these polynomials over the reals is ♯Phard for all points in R k but possibly for an exception set which is semialgebraic and of dimension strictly less than k. This result is analoguous to the characterization due to F. Jaeger, D. Vertigan and D. Welsh (1990) of the points where the Tutte polynomial is hard to evaluate. Our proof, however, builds mainly on the work by M. Dyer and C. Greenhill (2000). 1
Connection matrices and the definability of graph parameters
 In CSL
, 2009
"... In this paper we extend the Finite Rank Theorem for connection matrices of graph parameters definable in Monadic Second Order Logic with modular counting CMSOL of B. Godlin, T. Kotek and J.A. Makowsky, [16, 30], and demonstrate its vast applicability in simplifying known and new nondefinability res ..."
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In this paper we extend the Finite Rank Theorem for connection matrices of graph parameters definable in Monadic Second Order Logic with modular counting CMSOL of B. Godlin, T. Kotek and J.A. Makowsky, [16, 30], and demonstrate its vast applicability in simplifying known and new nondefinability results of graph properties and finding new nondefinability results for graph parameters. We also prove a FefermanVaught Theorem for the logic CFOL, First Order Logic with the modular counting quantifiers.