Delia Garijo

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 Erd?s and Szekeres, a polynomial time algorithm to compute distinguishing sets and determining sets of twin-free graphs, k-dominating sets, and matchings.

Polynomial graph invariants from homomorphism numbers

We give a new method of generating strongly polynomial sequences of graphs, i.e.,?sequences ( H k ) indexed by a tuple k = ( k 1 , ? , k h ) of positive integers, with the property that, for each fixed graph G , there is a multivariate polynomial p ( G ; x 1 , ? , x h ) such that the number of homomorphisms from G to H k is given by the evaluation p ( G ; k 1 , ? , k h ) . A classical example is the sequence of complete graphs ( K k ) , 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.

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 Bollobas, Pebody and Riordan.

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>=4 is realizable as the resolving number of an infinite family of graphs.

On Hamiltonian alternating cycles and paths

We undertake a study on computing Hamiltonian alternating cycles and paths on bicolored point sets. This has been an intensively studied problem, not always with a solution, when the paths and cycles are also required to be plane. In this paper, we relax the constraint on the cycles and paths from being plane to being 1-plane, and deal with the same type of questions as those for the plane case, obtaining a remarkable variety of results. Among them, we prove that a 1-plane Hamiltonian alternating cycle on a bicolored point set in general position can always be obtained, and that when the point set is in convex position, every Hamiltonian alternating cycle with minimum number of crossings is 1-plane. Further, for point sets in convex position, we provide

Stabbing Segments with Rectilinear Objects

O(n)

Stabbers of line segments in the plane

and

Shortcut sets for plane Euclidean networks (Extended abstract)

O(n^2)

Monochromatic geometric k-factors in red-blue sets with white and Steiner points

time algorithms for computing, respectively, Hamiltonian alternating cycles and paths with minimum number of crossings.

Graph homomorphisms, the Tutte polynomial and “q-state Potts uniqueness”

We consider stabbing regions for a set S of n line segments in the plane, that is, regions in the plane that contain exactly one endpoint of each segment of S. Concretely, we provide efficient algorithms for reporting all combinatorially different stabbing regions for S for regions that can be described as the intersection of axis-parallel halfplanes; these are halfplanes, strips, quadrants, 3-sided rectangles, and rectangles. The running times are O(n) (for the halfplane case), \(O(n\log n)\) (for strips, quadrants, and 3-sided rectangles), and \(O(n^2\log n)\) (for rectangles).