Reports de recerca
http://hdl.handle.net/2117/3198
2016-09-29T22:16:57ZOn the Partition Dimension and the Twin Number of a Graph
http://hdl.handle.net/2117/87267
On the Partition Dimension and the Twin Number of a Graph
Hernando Martín, María del Carmen; Mora Giné, Mercè; Pelayo Melero, Ignacio Manuel
A partition of the vertex set of a connected graph G is a locating partition of G if every vertex is uniquely determined by its vector of distances to the elements of . The partition dimension of G is the minimum cardinality of a locating partition of G . A pair of vertices u;v of a graph G are called twins if they have exactly the same set of neighbors other than u and v . A twin class is a maximal set of pairwise twin vertices. The twin number of a graph G is the maximum cardinality of a twin class of G . In this paper we undertake the study of the partition dimension of a graph by also considering its twin number. This approach allows us to obtain the set of connected graphs of order n 9 having partition dimension n
2016-05-24T10:33:18ZHernando Martín, María del CarmenMora Giné, MercèPelayo Melero, Ignacio ManuelA partition of the vertex set of a connected graph G is a locating partition of G if every vertex is uniquely determined by its vector of distances to the elements of . The partition dimension of G is the minimum cardinality of a locating partition of G . A pair of vertices u;v of a graph G are called twins if they have exactly the same set of neighbors other than u and v . A twin class is a maximal set of pairwise twin vertices. The twin number of a graph G is the maximum cardinality of a twin class of G . In this paper we undertake the study of the partition dimension of a graph by also considering its twin number. This approach allows us to obtain the set of connected graphs of order n 9 having partition dimension nOn cyclic Kautz digraphs
http://hdl.handle.net/2117/80848
On cyclic Kautz digraphs
Böhmová, Katerina; Dalfó Simó, Cristina; Huemer, Clemens
A prominent problem in Graph Theory is to find extremal graphs or digraphs with restrictions in their diameter, degree and number of vertices. Here we obtain a new family of digraphs with minimal diameter, that is, given the number of vertices and out-degree there is no other digraph with a smaller diameter. This new family is called modified cyclic digraphs MCK(d, `) and it is derived from the Kautz digraphs K(d, `). It is well-known that the Kautz digraphs K(d, `) have the smallest diameter among all digraphs with their number of vertices and degree. We define the cyclic Kautz digraphs
CK(d, `), whose vertices are labeled by all possible sequences a1 . . . a` of length `, such that each character ai is chosen from an alphabet containing d + 1 distinct symbols, where the consecutive characters in the sequence are different (as in Kautz digraphs), and now also requiring that a1 6= a`. The cyclic Kautz digraphs CK(d, `) have arcs between vertices a1a2 . . . a` and a2 . . . a`a`+1, with a1 6= a` and a2 6= a`+1. Unlike in Kautz digraphs K(d, `), any label of a vertex of CK(d, `) can be cyclically shifted to form again a label of a vertex of CK(d, `).
We give the main parameters of CK(d, `): number of vertices, number of arcs, and diameter.
Moreover, we construct the modified cyclic Kautz digraphs MCK(d, `) to obtain the same diameter as in the Kautz digraphs, and we show that MCK(d, `) are d-out-regular.
Finally, we compute the number of vertices of the iterated line digraphs of CK(d, `).
2015-12-17T10:58:24ZBöhmová, KaterinaDalfó Simó, CristinaHuemer, ClemensA prominent problem in Graph Theory is to find extremal graphs or digraphs with restrictions in their diameter, degree and number of vertices. Here we obtain a new family of digraphs with minimal diameter, that is, given the number of vertices and out-degree there is no other digraph with a smaller diameter. This new family is called modified cyclic digraphs MCK(d, `) and it is derived from the Kautz digraphs K(d, `). It is well-known that the Kautz digraphs K(d, `) have the smallest diameter among all digraphs with their number of vertices and degree. We define the cyclic Kautz digraphs
CK(d, `), whose vertices are labeled by all possible sequences a1 . . . a` of length `, such that each character ai is chosen from an alphabet containing d + 1 distinct symbols, where the consecutive characters in the sequence are different (as in Kautz digraphs), and now also requiring that a1 6= a`. The cyclic Kautz digraphs CK(d, `) have arcs between vertices a1a2 . . . a` and a2 . . . a`a`+1, with a1 6= a` and a2 6= a`+1. Unlike in Kautz digraphs K(d, `), any label of a vertex of CK(d, `) can be cyclically shifted to form again a label of a vertex of CK(d, `).
We give the main parameters of CK(d, `): number of vertices, number of arcs, and diameter.
Moreover, we construct the modified cyclic Kautz digraphs MCK(d, `) to obtain the same diameter as in the Kautz digraphs, and we show that MCK(d, `) are d-out-regular.
Finally, we compute the number of vertices of the iterated line digraphs of CK(d, `).Perfect anda quasiperfect domination in trees
http://hdl.handle.net/2117/77007
Perfect anda quasiperfect domination in trees
Cáceres, Jose; Hernando Martín, María del Carmen; Mora Giné, Mercè; Pelayo Melero, Ignacio Manuel; Puertas, M. Luz
2015-09-22T10:11:25ZCáceres, JoseHernando Martín, María del CarmenMora Giné, MercèPelayo Melero, Ignacio ManuelPuertas, M. LuzOn global location-domination in bipartite graphs
http://hdl.handle.net/2117/28318
On global location-domination in bipartite graphs
Hernando Martín, María del Carmen; Mora Giné, Mercè; Pelayo Melero, Ignacio Manuel
2015-06-16T09:42:50ZHernando Martín, María del CarmenMora Giné, MercèPelayo Melero, Ignacio ManuelOn perfect and quasiperfect domination in graphs
http://hdl.handle.net/2117/27709
On perfect and quasiperfect domination in graphs
Cáceres, José; Hernando Martín, María del Carmen; Mora Giné, Mercè; Pelayo Melero, Ignacio Manuel; Puertas, M Luz
2015-05-04T10:26:03ZCáceres, JoséHernando Martín, María del CarmenMora Giné, MercèPelayo Melero, Ignacio ManuelPuertas, M LuzGlobal location-domination in graphs
http://hdl.handle.net/2117/27680
Global location-domination in graphs
Hernando Martín, María del Carmen; Mora Giné, Mercè; Pelayo Melero, Ignacio Manuel
A dominating set S of a graph G is called
locating-dominating, LD-setfor short, if every vertex v not in S is uniquely determined by the set of neighbors of v
belonging to S. Locating-dominating sets of minimum cardinality are called LD-codes and the cardinality of an LD-code is the
location-domination number (G). An LD-set
S of a graph G is global if it is an LD-set of both G and its complement G. The
global location-domination number g(G) is the minimum cardinality of a global LD-set of
G. In this work,we give some relations between locating-dominating sets and the location-domination number in a graph and its complement
Domination, Global domination, Locating domination, Complement graph, Block-cactus, Trees
2015-04-30T08:12:31ZHernando Martín, María del CarmenMora Giné, MercèPelayo Melero, Ignacio ManuelA dominating set S of a graph G is called
locating-dominating, LD-setfor short, if every vertex v not in S is uniquely determined by the set of neighbors of v
belonging to S. Locating-dominating sets of minimum cardinality are called LD-codes and the cardinality of an LD-code is the
location-domination number (G). An LD-set
S of a graph G is global if it is an LD-set of both G and its complement G. The
global location-domination number g(G) is the minimum cardinality of a global LD-set of
G. In this work,we give some relations between locating-dominating sets and the location-domination number in a graph and its complementStrong product of graphs: Geodetic and hull number and boundary-type sets
http://hdl.handle.net/2117/8413
Strong product of graphs: Geodetic and hull number and boundary-type sets
Cáceres, José; Hernando Martín, María del Carmen; Mora Giné, Mercè; Pelayo Melero, Ignacio Manuel; Puertas González, María Luz
2010-07-27T09:58:31ZCáceres, JoséHernando Martín, María del CarmenMora Giné, MercèPelayo Melero, Ignacio ManuelPuertas González, María Luz4-labelings and grid embeddings of plane quadrangulations
http://hdl.handle.net/2117/3013
4-labelings and grid embeddings of plane quadrangulations
Barrière Figueroa, Eulalia; Huemer, Clemens
We show that each quadrangulation on $n$ vertices has a closed rectangle of influence drawing on the $(n-2) \times (n-2)$ grid.
Further, we present a simple algorithm to obtain a straight-line drawing of a quadrangulation on the
$\Big\lceil\frac{n}{2}\Big\rceil \times \Big\lceil\frac{3n}{4}\Big\rceil$ grid.
This is not optimal but has the advantage over other existing algorithms that it is not needed to add edges to
the quadrangulation to make it $4$-connected.
The algorithm is based on angle labeling and simple face counting in regions analogous to Schnyder's grid embedding for triangulation.
This extends previous results on book embeddings for quadrangulations from Felsner, Huemer, Kappes, and Orden (2008).
Our approach also yields a representation of a quadrangulation as a pair of rectangulations with a curious property.
2009-06-09T12:59:27ZBarrière Figueroa, EulaliaHuemer, ClemensWe show that each quadrangulation on $n$ vertices has a closed rectangle of influence drawing on the $(n-2) \times (n-2)$ grid.
Further, we present a simple algorithm to obtain a straight-line drawing of a quadrangulation on the
$\Big\lceil\frac{n}{2}\Big\rceil \times \Big\lceil\frac{3n}{4}\Big\rceil$ grid.
This is not optimal but has the advantage over other existing algorithms that it is not needed to add edges to
the quadrangulation to make it $4$-connected.
The algorithm is based on angle labeling and simple face counting in regions analogous to Schnyder's grid embedding for triangulation.
This extends previous results on book embeddings for quadrangulations from Felsner, Huemer, Kappes, and Orden (2008).
Our approach also yields a representation of a quadrangulation as a pair of rectangulations with a curious property.