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  • Nordhaus-Gaddum bounds for locating domination

     Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Pelayo Melero, Ignacio Manuel
    European journal of combinatorics
    Date of publication: 2014-02-01
    Journal article

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    A dominating set S of graph G is called metric-locating¿dominating if it is also locating, that is, if every vertex v is uniquely determined by its vector of distances to the vertices in S. If moreover, every vertex v not in S is also uniquely determined by the set of neighbors of v belonging to S, then it is said to be locating¿dominating. Locating, metric-locating¿dominating and locating¿dominating sets of minimum cardinality are called ß-codes, ¿-codes and ¿-codes, respectively. A Nordhaus¿Gaddum bound is a tight lower or upper bound on the sum or product of a parameter of a graph G and its complement View the MathML source. In this paper, we present some Nordhaus¿Gaddum bounds for the location number ß, the metric-location¿domination number ¿ and the location¿domination number ¿. Moreover, in each case, the graph family attaining the corresponding bound is fully characterized.

  • Locating-dominating codes: Bounds and extremal cardinalities

     Cáceres, Jose; Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Pelayo Melero, Ignacio Manuel; Puertas, M. Luz
    Applied mathematics and computation
    Date of publication: 2013-09-01
    Journal article

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    In this work, two types of codes such that they both dominate and locate the vertices of a graph are studied. Those codes might be sets of detectors in a network or processors controlling a system whose set of responses should determine a malfunctioning processor or an intruder. Here, we present our contributions on ¿-codes and ¿-codes concerning bounds, extremal values and realization theorems

  • Locating domination in graphs and their complements

     Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Pelayo Melero, Ignacio Manuel
    Date of publication: 2013-10-16
    Book chapter

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  • Nuevas cotas para la parametros de dominacion y localizacion en grafos

     Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Pelayo Melero, Ignacio Manuel
    Date of publication: 2013-10-16
    Book chapter

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  • Some structural, metric and convex properties of the boundary of a graph

     Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Pelayo Melero, Ignacio Manuel; Seara Ojea, Carlos
    Ars combinatoria
    Date of publication: 2013-04-25
    Journal article

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    Let u;v be two vertices of a connected graph G . The vertex v is said to be a boundary vertex of u if no neighbor of v is further away from u than v . The boundary of a graph is the set of all its boundary vertices. In this work, we present a number of properties of the boundary of a graph under diÆerent points of view: (1) a realization theorem involving diÆerent types of boundary vertex sets: extreme set, periphery, contour, and the whole boundary; (2) the contour is a monophonic set; and (3) the cardinality of the boundary is an upper bound for both the metric dimension and the determining number of a graph

  • Parámetros de localización y dominación de un grafo y su complementario

     Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Pelayo Melero, Ignacio Manuel
    Jornadas de Matemática Discreta y Algorítmica
    Presentation's date: 2012-07-11
    Presentation of work at congresses

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    Dado un grafo G = (V,E), un conjunto S ⊂ V es localiza-dominante si ϕ ̸= N(u)∩S ̸= N(v)∩S ̸= ϕ para todo u, v ∈ V \S. Notamos λ(G) al cardinal m´ınimo de estos conjuntos. En este trabajo estudiamos la relaci´on entre λ(G) y λ(G), damos cotas ajustadas de λ(G)+λ(G) y caracterizamos los grafos que alcanzan dichas cotas. En particular demostramos que λ(G) y λ(G) a lo sumo difieren en una unidad. En el caso particular de ser G un ´arbol y G ̸= P2 demostramos que λ(G) ≤ λ(G) ≤ λ(G)+1.

  • Watching systems in complete bipartite graphs

     Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Pelayo Melero, Ignacio Manuel
    Jornadas de Matemática Discreta y Algorítmica
    Presentation's date: 2012-07-11
    Presentation of work at congresses

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  • On the metric dimension of infinite graphs

     Cáceres, José; Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Pelayo Melero, Ignacio Manuel; Puertas, Maria Luz
    Discrete applied mathematics
    Date of publication: 2012-12-07
    Journal article

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    A set of vertices Sresolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of a graph G is the minimum cardinality of a resolving set. In this paper we study the metric dimension of infinite graphs such that all its vertices have finite degree. We give necessary conditions for those graphs to have finite metric dimension and characterize infinite trees with finite metric dimension. We also establish some results about the metric dimension of the cartesian product of finite and infinite graphs, and give the metric dimension of the cartesian product of several families of graphs.

    Infinite graph; Locally finite graph; Resolving set; Metric dimension; Cartesian product

  • Puntos y grafos: puentes geométricos (IP04 en CRP Comb. of points sets, ComPoSe,EuroGIGA ESF)

     Claverol Aguas, Merce; Dall, Aaron Matthew; Silveira, Rodrigo Ignacio; Huemer, Clemens; Mora Gine, Mercè; Sacristán Adinolfi, Vera; Hernando Martin, Maria Del Carmen; Seara Ojea, Carlos; Montes Lozano, Antonio; Hurtado Diaz, Fernando Alfredo
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    Boundary-type sets and product operators in graphs  Open access

     Cáceres, Jose; Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Pelayo Melero, Ignacio Manuel; Puertas, M. Luz
    Jornadas de Matemática Discreta y Algorítmica
    Presentation's date: 2010-07-07
    Presentation of work at congresses

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  • Valores extremos en los parámetros de dominación y resolución de un grafo

     Hernando Martin, Maria Del Carmen; Cáceres, Jose; Mora Gine, Mercè; Pelayo Melero, Ignacio Manuel; Puertas, M. Luz
    Jornadas de Matemática Discreta y Algorítmica
    Presentation's date: 2010-07
    Presentation of work at congresses

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  • Geodetic and hull numbers of strong products of graphs

     Hernando Martin, Maria Del Carmen; Cáceres, Jose; Mora Gine, Mercè; Pelayo Melero, Ignacio Manuel; Puertas, Maria Luz
    Jornadas de Matemática Discreta y Algorítmica
    Presentation's date: 2010-07
    Presentation of work at congresses

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    Extremal graph theory for metric dimension and diameter  Open access

     Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Pelayo Melero, Ignacio Manuel; Seara Ojea, Carlos; Wood, D. R.
    Electronic journal of combinatorics
    Date of publication: 2010
    Journal article

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    A set of vertices S resolves a connected graph G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of G is the minimum cardinality of a resolving set of G. Let G ,D be the set of graphs with metric dimension and diameter D. It is well-known that the minimum order of a graph in G ,D is exactly + D. The first contribution of this paper is to characterise the graphs in G ,D with order + D for all values of and D. Such a characterisation was previously only known for D 6 2 or 6 1. The second contribution is to determine the maximum order of a graph in G ,D for all values of D and . Only a weak upper bound was previously known.

  • PROBLEMAS DE COMBINATORIA Y DE COMPUTACION

     Claverol Aguas, Merce; Hernando Martin, Maria Del Carmen; Montes Lozano, Antonio; Mora Gine, Mercè; Sacristán Adinolfi, Vera; Seara Ojea, Carlos; Trias Pairo, Juan; Huemer, Clemens; Pfeifle, Julian Thoralf; Saumell Mendiola, Maria; Garcia Olaverri, Alfredo Martin; Tejel Altarriba, Francisco Javier; Dall, Aaron Matthew; Hurtado Diaz, Fernando Alfredo
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  • On the geodetic and the hull numbers in strong product graphs

     Cáceres, Jose; Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Pelayo Melero, Ignacio Manuel; Puertas, Maria Luz
    Computers & mathematics with applications
    Date of publication: 2010-12
    Journal article

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    A set S of vertices of a connected graph G is convex, if for any pair of vertices u,vS, every shortest path joining u and v is contained in S. The convex hull CH(S) of a set of vertices S is defined as the smallest convex set in G containing S. The set S is geodetic, if every vertex of G lies on some shortest path joining two vertices in S, and it is said to be a hull set if its convex hull is V(G). The geodetic and the hull numbers of G are the minimum cardinality of a geodetic and a minimum hull set, respectively. In this work, we investigate the behavior of both geodetic and hull sets with respect to the strong product operation for graphs. We also establish some bounds for the geodetic number and the hull number and obtain the exact value of these parameters for a number of strong product graphs.

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    On locating and dominating sets in graphs  Open access

     Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Cáceres, Jose; Puertas, Maria Luz; Pelayo Melero, Ignacio Manuel
    Workshop de Matemática Discreta Algarve/Andalucía. Encuentro Andaluz de Matemática Discreta
    Presentation's date: 2009-10-15
    Presentation of work at congresses

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  • On the metric dimension of infinite graphs

     Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Pelayo Melero, Ignacio Manuel
    Electronic notes in discrete mathematics
    Date of publication: 2009-11-04
    Journal article

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    A set of vertices S resolves a connected graph G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of a graph G is the minimum cardinality of a resolving set. In this paper we undertake the metric dimension of infinite locally finite graphs, i.e., those infinite graphs such that all its vertices have finite degree. We give some necessary conditions for an infinite graph to have finite metric dimension and characterize infinite trees with finite metric dimension. We also establish some general results about the metric dimension of the Cartesian product of finite and infinite graphs, and obtain the metric dimension of the Cartesian product of several families of graphs.

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    Strong product of graphs: Geodetic and hull number and boundary-type sets  Open access

     Cáceres, José; Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Pelayo Melero, Ignacio Manuel; Puertas, M. Luz
    Date: 2009-12
    Report

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    On some partitioning problems for two-colored point sets  Open access

     Grima, Clara; Hernando Martin, Maria Del Carmen; Huemer, Clemens; Hurtado Diaz, Fernando Alfredo
    Encuentros de Geometría Computacional
    Presentation's date: 2009
    Presentation of work at congresses

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    Let S be a two-colored set of n points in general position in the plane. We show that S admits at least 2 n 17 pairwise disjoint monochromatic triangles with vertices in S and empty of points of S. We further show that S can be partitioned into 3 n 11 subsets with pairwise disjoint convex hull such that within each subset all but at most one point have the same color. A lower bound on the number of subsets needed in any such partition is also given.

  • Geodeticity of the contour of chordal graphs

     Caceres, J; Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Pelayo Melero, Ignacio Manuel; Puertas, M L; Seara Ojea, Carlos
    Discrete applied mathematics
    Date of publication: 2008-04
    Journal article

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  • Fault-tolerant metric dimension of graphs

     Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Wood, David; Slater, P
    Date of publication: 2008-10
    Book chapter

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  • Complejidad de estructuras geométricas y combinatorias  Open access

     Hernando Martin, Maria Del Carmen
    Date of publication: 2008-01
    Book

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    En la presente memoria, se abordan cuatro problemas, existiendo en todos ellos una gran interacción entre la combinatoria y la geometría. El primer problema que se estudia es la introducción de varias extensiones del concepto de tipo de orden para nubes de puntos. Concretamente, se introducen los tipos de orden circulares y triángulares, en las versiones orientada y no orientada. Se han demostrado resultados combinatorios análogos a resultados bien conocidos sobre tipos de orden ordinarios, introducidos por Goodman y Pollack como es el llamado Teorema de ordenación geométrica. Se ha estudiado también la información geométrica que proporciona cada uno de estos conceptos. El segundo problema estudia el empaquetamiento plano de grafos; esto es, el trazado de grafos, disjuntos en aristas, en el plano. Hemos obtenido varios resultados sobre el empaquetamiento plano de árboles y ciclos. Concretamente, para árboles que no sean estrellas, se ha demostrado que siempre admiten empaquetamiento plano: dos copias de un árbol cualquiera, un árbol cualquiera y un camino, un árbol cualquiera y un ciclo. También se han obtenido resultados sobre empaquetamiento plano de dos o tres ciclos. La principal herramienta que se ha utilizado es la representación de un árbol en un polígono convexo con propiedades muy concretas. En tercer lugar se estudia el grafo T (P) de árboles geométricos de una nube de puntos P, siendo este grafo el que tiene por vértices los árboles generadores sin cortes de P y dos de tales árboles T1, T2 son aduacentes si y sólo s, T2C=t1e+f para ciertas aristas e y f. Se han obtenido propiedades combinatorias de estos grafos, especialmente en el caso particular en que el conjunto de puntos esta en posición convexa. En este caso se ha determinado el centro, radio y grupo de automofismos de estos grafos, y demostrado que son hamiltonianos y de conectividad máxima. Finalmente, también se ha estudiado el grafo Mm de los emparejamientos perfectos sin cortes de una nube de 2m puntos en posición convexa. Entre los resultados obtenidos cabe destacar que se ha demostrado que Mm es bipartito, hamiltoniano sólo si m es par y que el diámetro de Mm es igual a m-1, siendo todos los emparejamientos de excentricidad máxima.

  • On the metric dimension of infinite graphs

     Cáceres, J; Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Pelayo Melero, Ignacio Manuel; Puertas, M L
    Jornadas de Matemática Discreta y Algorítmica
    Presentation of work at congresses

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  • Producto fuerte de grafos: invariantes de convexidad y conjuntos fronterizos

     Cáceres, José; Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Pelayo Melero, Ignacio Manuel; Puertas, María L
    Jornadas de Matemática Discreta y Algorítmica
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  • Dimensión métrica tolerante de un grafo

     Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Slater, Peter J; Wood, David
    Jornadas de Matemática Discreta y Algorítmica
    Presentation of work at congresses

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  • Extremal Graph Theory for Metric Dimension and Diameter

     Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Pelayo Melero, Ignacio Manuel; Seara Ojea, Carlos; Wood, D R
    Electronic notes in discrete mathematics
    Date of publication: 2007-08
    Journal article

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  • On the Metric Dimension of Cartesian Products of Graphs

     Cáceres, J; Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Pelayo Melero, Ignacio Manuel; Puertas, M L; Seara Ojea, Carlos; Wood, David
    SIAM journal on discrete mathematics
    Date of publication: 2007-05
    Journal article

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  • Dimensión métrica de grafos infinitos

     Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Seara Ojea, Carlos; Cáceres, J; Moreno-Gonzalez, A; Pelayo Melero, Ignacio Manuel; Puertas, M L
    Date of publication: 2007-03
    Book chapter

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  • Dimensión métrica de grafos infinitos

     Cáceres, J; Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Moreno, A; Pelayo Melero, Ignacio Manuel; Puertas, M L; Seara Ojea, Carlos
    Encuentro Andaluz de Matemática Discreta
    Presentation's date: 2007
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  • Grafos de orden máximo y mínimo con diàmetro y dimensión métrica fijados

     Seara Ojea, Carlos; Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Pelayo Melero, Ignacio Manuel; Wood, D R
    Jornadas de Matemática Discreta y Algorítmica
    Presentation's date: 2007-07-13
    Presentation of work at congresses

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  • Extremal Graph Theory for Metric Dimension and Diameter

     Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Pelayo Melero, Ignacio Manuel; Seara Ojea, Carlos; Wood, David
    European Conference on Combinatorics, Graph Theory, and Applications
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  • On geodetic sets formed by boundary vertices

     Cáceres, J; Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Pelayo Melero, Ignacio Manuel; Puertas, M L; Seara Ojea, Carlos
    Discrete mathematics
    Date of publication: 2006-02
    Journal article

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  • Some structural, metric and convex properties on the boundary of a graph

     Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Pelayo Melero, Ignacio Manuel; Seara Ojea, Carlos
    Electronic notes in discrete mathematics
    Date of publication: 2006-07
    Journal article

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  • Grafos de orden máximo y mínimo con diámetro y dimensión métrica fijados

     Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Pelayo Melero, Ignacio Manuel; Seara Ojea, Carlos; Wood, D R
    Date of publication: 2006-01
    Book chapter

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  • On the metric dimension of cartesian products of graphs

     Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Pelayo Melero, Ignacio Manuel; Seara Ojea, Carlos; Cáceres, J; Puertas, M L; Wood, D
    Date of publication: 2006-01
    Book chapter

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  • METODOS ALGORITMICOS Y ESTRUCTURAS COMBINATORIAS EN GEOMETRIA COMPUTACIONAL

     Hernando Martin, Maria Del Carmen; Hurtado Diaz, Fernando Alfredo; Montes Lozano, Antonio; Sacristán Adinolfi, Vera; Mora Gine, Mercè
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  • On the metric dimension of cartesian products of graphs

     Cáceres, J; Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Pelayo Melero, Ignacio Manuel; Seara Ojea, Carlos; Wood, D R
    Jornadas de Matemática Discreta y Algorítmica
    Presentation's date: 2006-07-13
    Presentation of work at congresses

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  • On the metric dimension of some families of graphs

     Cáceres, J; Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Pelayo Melero, Ignacio Manuel; Seara Ojea, Carlos
    Electronic notes in discrete mathematics
    Date of publication: 2005-10
    Journal article

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  • Searching for geodetic boundary vertex sets

     Cáceres, J; Puertas, M L; Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Pelayo Melero, Ignacio Manuel; Seara Ojea, Carlos
    Electronic notes in discrete mathematics
    Date of publication: 2005-01
    Journal article

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  • On the Steiner, geodetic and hull number of graphs

     Hernando Martin, Maria Del Carmen; Jiang, T; Mora Gine, Mercè; Pelayo Melero, Ignacio Manuel; Seara Ojea, Carlos
    Discrete mathematics
    Date of publication: 2005-04
    Journal article

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  • El dígrafo excéntrico de un grafo intervalo

     Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Pelayo Melero, Ignacio Manuel; Seara Ojea, Carlos; Cáceres, J; Puertas, M L
    Date of publication: 2005-06
    Book chapter

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  • Reconstrucción de un grafo a partir de la clausura geodética

     Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Pelayo Melero, Ignacio Manuel; Seara Ojea, Carlos; Cáceres, J; Puertas, M L
    Date of publication: 2005-06
    Book chapter

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  • Eines bàsiques d'àlgebra lineal

     Hernando Martin, Maria Del Carmen; Magret Planas, Maria Dels Dolors; Puig Pla, Carles Maria; Massa Esteve, Maria Rosa
    Date of publication: 2005-10
    Book

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  • On the metric dimension of some families of graphs

     Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Pelayo Melero, Ignacio Manuel; Seara Ojea, Carlos; Cáceres, J; Puertas, M L
    7th International Colloquium on Graph Theory
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  • Grupo de investigación consolidado de la Generalitat de Catalunya

     Hurtado Diaz, Fernando Alfredo; Hernando Martin, Maria Del Carmen; Montes Lozano, Antonio; Trias Pairo, Juan; Sacristán Adinolfi, Vera; Mora Gine, Mercè
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  • Searching for geodesic boundary vertex set

     Cáceres, J; Puertas, M L; Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Pelayo Melero, Ignacio Manuel; Seara Ojea, Carlos
    2nd Brazilian Symposium on Graphs, Algorithms, and Combinatorics
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  • El diagrafo excéntrico de un grafo intervalo

     Cáceres, J; Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Pelayo Melero, Ignacio Manuel; Puertas, M L; Seara Ojea, Carlos
    Encuentros de Geometría Computacional
    Presentation's date: 2005
    Presentation of work at congresses

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  • Reconstrucción de un grafo a partir de la clausura geodética

     Mora Gine, Mercè; Seara Ojea, Carlos; Hernando Martin, Maria Del Carmen; Pelayo Melero, Ignacio Manuel
    Encuentros de Geometría Computacional
    Presentation's date: 2005
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  • Estudio del contorno en grafos cordales

     Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Cáceres, J; Pelayo Melero, Ignacio Manuel; Puertas, M L; Seara Ojea, Carlos
    Date of publication: 2004-09
    Book chapter

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  • On geodesic and monophonic convexity

     Hernando Martin, Maria Del Carmen; Mora Gine, Mercè; Pelayo Melero, Ignacio Manuel; Seara Ojea, Carlos
    Date of publication: 2004-01
    Book chapter

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