Reports de recerca
http://hdl.handle.net/2117/3198
2018-01-24T03:14:08ZLocating domination in bipartite graphs and their complements
http://hdl.handle.net/2117/111067
Locating domination in bipartite graphs and their complements
Hernando Martín, María del Carmen; Mora Giné, Mercè; Pelayo Melero, Ignacio Manuel
A set S of vertices of a graph G is distinguishing if the sets of neighbors in S for every pair of vertices not in S are distinct. A locating-dominating set of G is a dominating distinguishing set. The location-domination number of G , ¿ ( G ), is the minimum cardinality of a locating-dominating set. In this work we study relationships between ¿ ( G ) and ¿ ( G ) for bipartite graphs. The main result is the characterization of all connected bipartite graphs G satisfying ¿ ( G ) = ¿ ( G ) + 1. To this aim, we define an edge-labeled graph G S associated with a distinguishing set S that turns out to be very helpful
2017-11-22T12:08:39ZHernando Martín, María del CarmenMora Giné, MercèPelayo Melero, Ignacio ManuelA set S of vertices of a graph G is distinguishing if the sets of neighbors in S for every pair of vertices not in S are distinct. A locating-dominating set of G is a dominating distinguishing set. The location-domination number of G , ¿ ( G ), is the minimum cardinality of a locating-dominating set. In this work we study relationships between ¿ ( G ) and ¿ ( G ) for bipartite graphs. The main result is the characterization of all connected bipartite graphs G satisfying ¿ ( G ) = ¿ ( G ) + 1. To this aim, we define an edge-labeled graph G S associated with a distinguishing set S that turns out to be very helpfulMetric-locating-dominating partitions in graphs
http://hdl.handle.net/2117/111061
Metric-locating-dominating partitions in graphs
Hernando Martín, María del Carmen; Mora Giné, Mercè; Pelayo Melero, Ignacio Manuel
A partition ¿ = { S 1 ,...,S k } of the vertex set of a connected graph G is a metric-locating partition of G if for every pair of vertices u,v belonging to the same part S i , d ( u,S j ) 6 = d ( v,S j ), for some other part S j . The partition dimension ß p ( G ) is the minimum cardinality of a metric- locating partition of G . A metric-locating partition ¿ is called metric-locating-dominanting if for every vertex v of G , d ( v,S j ) = 1, for some part S j of ¿. The partition metric-location-domination number ¿ p ( G ) is the minimum cardinality of a metric-locating-dominating partition of G . In this paper we show, among other results, that ß p ( G ) = ¿ p ( G ) = ß p ( G ) + 1. We also charac- terize all connected graphs of order n = 7 satisfying any of the following conditions: ¿ p ( G ) = n - 1, ¿ p ( G ) = n - 2 and ß p ( G ) = n - 2. Finally, we present some tight Nordhaus-Gaddum bounds for both the partition dimension ß ( G ) and the partition metric-location-domination number ¿ ( G ). Keywords: dominating partition, locating partition, location, domination, metric location
2017-11-22T11:22:19ZHernando Martín, María del CarmenMora Giné, MercèPelayo Melero, Ignacio ManuelA partition ¿ = { S 1 ,...,S k } of the vertex set of a connected graph G is a metric-locating partition of G if for every pair of vertices u,v belonging to the same part S i , d ( u,S j ) 6 = d ( v,S j ), for some other part S j . The partition dimension ß p ( G ) is the minimum cardinality of a metric- locating partition of G . A metric-locating partition ¿ is called metric-locating-dominanting if for every vertex v of G , d ( v,S j ) = 1, for some part S j of ¿. The partition metric-location-domination number ¿ p ( G ) is the minimum cardinality of a metric-locating-dominating partition of G . In this paper we show, among other results, that ß p ( G ) = ¿ p ( G ) = ß p ( G ) + 1. We also charac- terize all connected graphs of order n = 7 satisfying any of the following conditions: ¿ p ( G ) = n - 1, ¿ p ( G ) = n - 2 and ß p ( G ) = n - 2. Finally, we present some tight Nordhaus-Gaddum bounds for both the partition dimension ß ( G ) and the partition metric-location-domination number ¿ ( G ). Keywords: dominating partition, locating partition, location, domination, metric locationDominating 2- broadcast in graphs: complexity, bounds and extremal graphs
http://hdl.handle.net/2117/109101
Dominating 2- broadcast in graphs: complexity, bounds and extremal graphs
Hernando Martín, María del Carmen; Mora Giné, Mercè; Pelayo Melero, Ignacio Manuel; Cáceres, José; Puertas, M. Luz
Limited dominating broadcasts were proposed as a variant of dominating broadcasts, where the broadcast function is upper bounded. As a natural extension of domination, we consider dominating 2-broadcasts along with the associated parameter, the dominating 2-broadcast number. We prove that computing the dominating 2-broadcast number is a NP-complete problem, but can be achieved in linear time for trees. We also give an upper bound for this parameter, that is tight for graphs as large as desired
2017-10-25T05:28:42ZHernando Martín, María del CarmenMora Giné, MercèPelayo Melero, Ignacio ManuelCáceres, JoséPuertas, M. LuzLimited dominating broadcasts were proposed as a variant of dominating broadcasts, where the broadcast function is upper bounded. As a natural extension of domination, we consider dominating 2-broadcasts along with the associated parameter, the dominating 2-broadcast number. We prove that computing the dominating 2-broadcast number is a NP-complete problem, but can be achieved in linear time for trees. We also give an upper bound for this parameter, that is tight for graphs as large as desiredThe Plogi and ACi-1 operators on the polynomial time hierarchy
http://hdl.handle.net/2117/97278
The Plogi and ACi-1 operators on the polynomial time hierarchy
Castro Rabal, Jorge; Seara Ojea, Carlos
In a previous paper ([CS-92]) we studied the agreement of operators P_{log^i} and AC^{i-1} acting on NP. In this article we extend this work to other classes of the polynomial time hierarchy. We show that on Sigma_k^p, Pi_k^p, Delta_k^P and Theta_k^P-classes both operators have the same behaviour, but this coincidence does not seem to be true on other classes included in the PH hierarchy: we give a set A such that, relativized to A, P_{log^i}(P_{log^j}(NP)) is different from AC^{i-1}(P_{log^j}(NP)). As a result of these characterizations we show P_{log}(Theta_k^p) = Theta_k^p, an equality that is useful to show lowness properties. In fact, we get easily the Theta-lowness results given by Long and Sheu in their paper [LS-91]. Besides, we clarify the situation of the classes in L_2^{p,Delta} for which their membership to L_2^{p,Theta} was not clear.
2016-11-28T09:01:43ZCastro Rabal, JorgeSeara Ojea, CarlosIn a previous paper ([CS-92]) we studied the agreement of operators P_{log^i} and AC^{i-1} acting on NP. In this article we extend this work to other classes of the polynomial time hierarchy. We show that on Sigma_k^p, Pi_k^p, Delta_k^P and Theta_k^P-classes both operators have the same behaviour, but this coincidence does not seem to be true on other classes included in the PH hierarchy: we give a set A such that, relativized to A, P_{log^i}(P_{log^j}(NP)) is different from AC^{i-1}(P_{log^j}(NP)). As a result of these characterizations we show P_{log}(Theta_k^p) = Theta_k^p, an equality that is useful to show lowness properties. In fact, we get easily the Theta-lowness results given by Long and Sheu in their paper [LS-91]. Besides, we clarify the situation of the classes in L_2^{p,Delta} for which their membership to L_2^{p,Theta} was not clear.General bounds on limited broadcast domination
http://hdl.handle.net/2117/96711
General bounds on limited broadcast domination
Hernando Martín, María del Carmen; Mora Giné, Mercè; Pelayo Melero, Ignacio Manuel; Puertas, Maria Luz; Cáceres, José
Limited dominating broadcasts were proposed as a variant of dominating broadcasts, where the broadcast function is upper bounded by a constant k . The minimum cost of such a dominating broadcast is the k -broadcast dominating number. We present a uni ed upper bound on this parameter for any value of k in terms of both k and the order of the graph. For the speci c case of the 2-broadcast dominating number, we show that this bound is tight for graphs as large as desired. We also study the family of caterpillars, providing a smaller upper bound, which is attained by a set of such graphs with unbounded order.
2016-11-16T10:37:47ZHernando Martín, María del CarmenMora Giné, MercèPelayo Melero, Ignacio ManuelPuertas, Maria LuzCáceres, JoséLimited dominating broadcasts were proposed as a variant of dominating broadcasts, where the broadcast function is upper bounded by a constant k . The minimum cost of such a dominating broadcast is the k -broadcast dominating number. We present a uni ed upper bound on this parameter for any value of k in terms of both k and the order of the graph. For the speci c case of the 2-broadcast dominating number, we show that this bound is tight for graphs as large as desired. We also study the family of caterpillars, providing a smaller upper bound, which is attained by a set of such graphs with unbounded order.New results on metric-locating-dominating sets of graphs
http://hdl.handle.net/2117/96709
New results on metric-locating-dominating sets of graphs
Hernando Martín, María del Carmen; Mora Giné, Mercè; González, Antonio
A dominating set S of a graph is a metric-locating-dominating set if each vertex of the graph is uniquely distinguished by its distanc es from the elements of S , and the minimum cardinality of such a set is called the metri c-location- domination number. In this paper, we undertake a study that, in general graphs and specific families, relates metric-locating-dominatin g sets to other special sets: resolving sets, dominating sets, locating-dominating set s and doubly resolving sets. We first characterize classes of trees according to cer tain relationships between their metric-location-domination number and thei r metric dimension and domination number. Then, we show different methods to tran sform metric- locating-dominating sets into locating-dominating sets a nd doubly resolving sets. Our methods produce new bounds on the minimum cardinalities of all those sets, some of them involving parameters that have not been related so far
2016-11-16T10:25:27ZHernando Martín, María del CarmenMora Giné, MercèGonzález, AntonioA dominating set S of a graph is a metric-locating-dominating set if each vertex of the graph is uniquely distinguished by its distanc es from the elements of S , and the minimum cardinality of such a set is called the metri c-location- domination number. In this paper, we undertake a study that, in general graphs and specific families, relates metric-locating-dominatin g sets to other special sets: resolving sets, dominating sets, locating-dominating set s and doubly resolving sets. We first characterize classes of trees according to cer tain relationships between their metric-location-domination number and thei r metric dimension and domination number. Then, we show different methods to tran sform metric- locating-dominating sets into locating-dominating sets a nd doubly resolving sets. Our methods produce new bounds on the minimum cardinalities of all those sets, some of them involving parameters that have not been related so farOn 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 Manuel