Reports de recerca
http://hdl.handle.net/2117/3180
2017-07-23T06:59:04ZFrom subKautz digraphs to cyclic Kautz digraphs
http://hdl.handle.net/2117/105173
From subKautz digraphs to cyclic Kautz digraphs
Dalfó Simó, Cristina
Kautz digraphs K(d,l) are a well-known family of dense digraphs, widely studied as a good model for interconnection networks. Closely related with these, the cyclic Kautz digraphs CK(d,l) were recently introduced by Böhmová, Huemer and the author, and some of its distance-related parameters were fixed. In this paper we propose a new approach to cyclic Kautz digraphs by introducing the family of subKautz digraphs sK(d,l), from where the cyclic Kautz digraphs can be obtained as line digraphs. This allows us to give exact formulas for the distance between any two vertices of both sK(d,l) and CK(d,l). Moreover, we compute the diameter and the semigirth of both families, also providing efficient routing algorithms to find the shortest path between any pair of vertices. Using these parameters, we also prove that sK(d,l) and CK(d,l) are maximally vertex-connected and super-edge-connected. Whereas K(d,l) are optimal with respect to the diameter, we show that sK(d,l) and CK(d,l) are optimal with respect to the mean distance, whose exact values are given for both families when l = 3. Finally, we provide a lower bound on the girth of CK(d,l) and sK(d,l)
2017-06-06T12:12:17ZDalfó Simó, CristinaKautz digraphs K(d,l) are a well-known family of dense digraphs, widely studied as a good model for interconnection networks. Closely related with these, the cyclic Kautz digraphs CK(d,l) were recently introduced by Böhmová, Huemer and the author, and some of its distance-related parameters were fixed. In this paper we propose a new approach to cyclic Kautz digraphs by introducing the family of subKautz digraphs sK(d,l), from where the cyclic Kautz digraphs can be obtained as line digraphs. This allows us to give exact formulas for the distance between any two vertices of both sK(d,l) and CK(d,l). Moreover, we compute the diameter and the semigirth of both families, also providing efficient routing algorithms to find the shortest path between any pair of vertices. Using these parameters, we also prove that sK(d,l) and CK(d,l) are maximally vertex-connected and super-edge-connected. Whereas K(d,l) are optimal with respect to the diameter, we show that sK(d,l) and CK(d,l) are optimal with respect to the mean distance, whose exact values are given for both families when l = 3. Finally, we provide a lower bound on the girth of CK(d,l) and sK(d,l)On quotient digraphs and voltage digraphs
http://hdl.handle.net/2117/105171
On quotient digraphs and voltage digraphs
Dalfó Simó, Cristina; Fiol Mora, Miquel Àngel; Miller, Mirka; Ryan, Joe
In this note we present a general approach to construct large digraphs from small ones. These are called expanded digraphs, and, as particular cases, we show their close relationship between voltage digraphs and line digraphs, which are two known approaches to obtain dense digraphs. In the same context, we show the equivalence
between the vertex-splitting and partial line digraph techniques. Then, we give a sufficient condition for a lifted digraph of a base line digraph to be again a line digraph. Some of the results are illustrated with two well-known families of digraphs. Namely, De Bruijn and Kautz digraphs.
2017-06-06T12:00:49ZDalfó Simó, CristinaFiol Mora, Miquel ÀngelMiller, MirkaRyan, JoeIn this note we present a general approach to construct large digraphs from small ones. These are called expanded digraphs, and, as particular cases, we show their close relationship between voltage digraphs and line digraphs, which are two known approaches to obtain dense digraphs. In the same context, we show the equivalence
between the vertex-splitting and partial line digraph techniques. Then, we give a sufficient condition for a lifted digraph of a base line digraph to be again a line digraph. Some of the results are illustrated with two well-known families of digraphs. Namely, De Bruijn and Kautz digraphs.Connected and internal graph searching
http://hdl.handle.net/2117/97422
Connected and internal graph searching
Barrière Figueroa, Eulalia; Fraigniaud, Pierre; Santoro, Nicola; Thilikos Touloupas, Dimitrios
This paper is concerned with the graph searching game. The search number es(G) of a graph G is the smallest number of searchers required to clear G. A search strategy is monotone (m) if no recontamination ever occurs. It is connected (c) if the set of clear edges always forms a connected subgraph. It is internal (i) if the removal of searchers is not allowed. The difficulty of the connected version and of the monotone internal version of the graph searching problem comes from the fact that, as shown in the paper, none of these problems is minor closed for arbitrary graphs, as opposed to all known variants of the graph searching problem. Motivated by the fact that connected graph searching, and monotone internal graph searching are both minor closed in trees, we provide a complete characterization of the set of trees that can be cleared by a given number of searchers. In fact, we show that, in trees, there is only one obstruction for monotone internal search, as well as for connected search, and this obstruction is the same for the two problems. This allows us to prove that, for any tree T, mis(T)= cs(T). For arbitrary graphs, we prove that there is a unique chain of inequalities linking all the search numbers above. More precisely, for any graph G, es(G)= is(G)= ms(G)leq mis(G)leq cs(G)= ics(G)leq mcs(G)=mics(G). The first two inequalities can be strict. In the case of trees, we have mics(G)leq 2 es(T)-2, that is there are exactly 2 different search numbers in trees, and these search numbers differ by a factor of 2 at most.
2016-11-29T13:30:39ZBarrière Figueroa, EulaliaFraigniaud, PierreSantoro, NicolaThilikos Touloupas, DimitriosThis paper is concerned with the graph searching game. The search number es(G) of a graph G is the smallest number of searchers required to clear G. A search strategy is monotone (m) if no recontamination ever occurs. It is connected (c) if the set of clear edges always forms a connected subgraph. It is internal (i) if the removal of searchers is not allowed. The difficulty of the connected version and of the monotone internal version of the graph searching problem comes from the fact that, as shown in the paper, none of these problems is minor closed for arbitrary graphs, as opposed to all known variants of the graph searching problem. Motivated by the fact that connected graph searching, and monotone internal graph searching are both minor closed in trees, we provide a complete characterization of the set of trees that can be cleared by a given number of searchers. In fact, we show that, in trees, there is only one obstruction for monotone internal search, as well as for connected search, and this obstruction is the same for the two problems. This allows us to prove that, for any tree T, mis(T)= cs(T). For arbitrary graphs, we prove that there is a unique chain of inequalities linking all the search numbers above. More precisely, for any graph G, es(G)= is(G)= ms(G)leq mis(G)leq cs(G)= ics(G)leq mcs(G)=mics(G). The first two inequalities can be strict. In the case of trees, we have mics(G)leq 2 es(T)-2, that is there are exactly 2 different search numbers in trees, and these search numbers differ by a factor of 2 at most.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.Symmetry breaking in tournaments
http://hdl.handle.net/2117/91158
Symmetry breaking in tournaments
Lozano Bojados, Antoni
We provide upper bounds for the determining number and the metric dimension of tournaments. A set of vertices S is a determining set for a tournament T if every nontrivial automorphism of T moves at least one vertex of S, while S is a resolving set for T if every two distinct vertices in T have different distances to some vertex in S. We show that the minimum size of a determining set for an order n tournament (its determining number) is bounded by n/3, while the minimum size of a resolving set for an order n strong tournament (its metric dimension) is bounded by n/2. Both bounds are optimal.
2016-10-27T11:29:19ZLozano Bojados, AntoniWe provide upper bounds for the determining number and the metric dimension of tournaments. A set of vertices S is a determining set for a tournament T if every nontrivial automorphism of T moves at least one vertex of S, while S is a resolving set for T if every two distinct vertices in T have different distances to some vertex in S. We show that the minimum size of a determining set for an order n tournament (its determining number) is bounded by n/3, while the minimum size of a resolving set for an order n strong tournament (its metric dimension) is bounded by n/2. Both bounds are optimal.On 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 ManuelApproximate results for rainbow labelings
http://hdl.handle.net/2117/27843
Approximate results for rainbow labelings
Lladó Sánchez, Ana M.
Article de recerca
2015-05-08T11:21:05ZLladó Sánchez, Ana M.