75 research outputs found
Graph homomorphisms, the Tutte polynomial and “q-state Potts uniqueness”
We establish for which weighted graphs H homomorphism functions from multigraphs
G to H are specializations of the Tutte polynomial of G, answering a question
of Freedman, Lov´asz and Schrijver.
We introduce a new property of graphs called “q-state Potts uniqueness” and relate
it to chromatic and Tutte uniqueness, and also to “chromatic–flow uniqueness”,
recently studied by Duan, Wu and Yu.Ministerio de Educación y Ciencia MTM2005-08441-C02-0
Distinguishing graphs by their left and right homomorphism profiles
We introduce a new property of graphs called ‘q-state Potts unique-ness’ and relate it to chromatic and Tutte
uniqueness, and also to ‘chromatic–flow uniqueness’, recently studied by Duan, Wu and Yu.
We establish for which edge-weighted graphs H homomor-phism functions from multigraphs G to H are
specializations of the Tutte polynomial of G, in particular answering a question of Freed-man, Lovász and
Schrijver. We also determine for which edge-weighted graphs H homomorphism functions from
multigraphs G to H are specializations of the ‘edge elimination polynomial’ of Averbouch, Godlin and
Makowsky and the ‘induced subgraph poly-nomial’ of Tittmann, Averbouch and Makowsky.
Unifying the study of these and related problems is the notion of the left and right homomorphism profiles
of a graph.Ministerio de Educación y Ciencia MTM2008-05866-C03-01Junta de Andalucía FQM- 0164Junta de Andalucía P06-FQM-0164
Polynomial graph invariants from homomorphism numbers
We give a new method of generating strongly polynomial sequences of graphs, i.e., sequences
(Hk) indexed by a tuple k = (k1, . . . , kh) of positive integers, with the property
that, for each fixed graph G, there is a multivariate polynomial p(G; x1, . . . , xh) such that
the number of homomorphisms from G to Hk is given by the evaluation p(G; k1, . . . , kh).
A classical example is the sequence of complete graphs (Kk), for which p(G; x) is the chromatic
polynomial of G. Our construction is based on tree model representations of graphs. It
produces a large family of graph polynomials which includes the Tutte polynomial,
the Averbouch–Godlin–Makowsky polynomial, and the Tittmann–Averbouch–Makowsky
polynomial. We also introduce a new graph parameter, the branching core size of a simple
graph, derived from its representation under a particular tree model, and related to
how many involutive automorphisms it has. We prove that a countable family of graphs of
bounded branching core size is always contained in the union of a finite number of strongly
polynomial sequences.Ministerio de Economía y Competitividad MTM2014-60127-
On the number of B-flows of a graph
We exhibit explicit constructions of contractors for the graph parameter counting the number of B-flows
of a graph, where B is a subset of a finite Abelian group closed under inverses. These constructions are of
great interest because of their relevance to the family of B-flow conjectures formulated by Tutte, Fulkerson,
Jaeger, and others.Junta de Andalucía FQM-016
The difference between the metric dimension and the determining number of a graph
We study the maximum value of the difference between the metric dimension and the determining number of a graph as a function of its order. We develop a technique that uses functions related to locating-dominating sets to obtain lower and upper bounds on that maximum, and exact computations when restricting to some specific families of graphs. Our approach requires very diverse tools and connections with well-known objects in graph theory; among them: a classical result in graph domination by Ore, a Ramsey-type result by Erdős and Szekeres, a polynomial time algorithm to compute distinguishing sets and determining sets of twin-free graphs, k-dominating sets, and matchings
Homomorphisms and polynomial invariants of graphs
This paper initiates a general study of the connection between graph homomorphisms and the Tutte
polynomial. This connection can be extended to other polynomial invariants of graphs related to the Tutte
polynomial such as the transition, the circuit partition, the boundary, and the coboundary polynomials.
As an application, we describe in terms of homomorphism counting some fundamental evaluations of the
Tutte polynomial in abelian groups and statistical physics. We conclude the paper by providing a
homomorphism view of the uniqueness conjectures formulated by Bollobás, Pebody and Riordan.Ministerio de Educación y Ciencia MTM2005-08441-C02-01Junta de Andalucía PAI-FQM-0164Junta de Andalucía P06-FQM-0164
On the metric dimension, the upper dimension and the resolving number of graphs
This paper deals with three resolving parameters: the metric dimension, the upper dimension and the resolving number. We first answer a question raised by Chartrand and Zhang asking for a characterization of the graphs with equal metric dimension and resolving number. We also solve in the affirmative a conjecture posed by Chartrand, Poisson and Zhang about the realization of the metric dimension and the upper dimension. Finally, we prove that no integer a≥4a≥4 is realizable as the resolving number of an infinite family of graphs
Tutte uniqueness of locally grid graphs
A graph is said to be locally grid if the structure around each of
its vertices is a 3 × 3 grid. As a follow up of the research initiated
in [4] and [3] we prove that most locally grid graphs are uniquely
determined by their Tutte polynomial.Ministerio de Ciencia y Tecnología BFM2001-2474-ORIJunta de Andalucía PAI FQM-16
Computing optimal shortcuts for networks
We augment a plane Euclidean network with a segment or shortcut to minimize the largest distance between any two points along the edges of the resulting network. In this continuous setting, the problem of computing distances and placing a shortcut is much harder as all points on the network, instead of only the vertices, must be taken into account. Our main result for general networks states that it is always possible to determine in polynomial time whether the network has an optimal shortcut and compute one in case of existence. We also improve this general method for networks that are paths, restricted to using two types of shortcuts: those of any fixed direction and shortcuts that intersect the path only on its endpoints.Peer ReviewedPostprint (published version
Computing optimal shortcuts for networks
We study augmenting a plane Euclidean network with a segment, called shortcut, to minimize the largest distance between any two points along the edges of the resulting network. Questions of this type have received considerable attention recently, mostly for discrete variants of the problem. We study a fully continuous setting, where all points on the network and the inserted segment must be taken into account. We present the first results on the computation of optimal shortcuts for general networks in this model, together with several results for networks that are paths, restricted to two types of shortcuts: shortcuts with a fixed orientation and simple shortcuts.Peer ReviewedPostprint (published version
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