Reports de recerca
http://hdl.handle.net/2117/3180
2024-03-29T11:48:00ZContiguous and internal graph searching
http://hdl.handle.net/2117/369736
Contiguous and internal graph searching
Barrière Figueroa, Eulalia; Fraigniaud, Pierre; Santoro, Nicola; Thilikos, Dimitrios M.
In the graph searching problem, we are given a graph whose edges are all "contaminated", and, via a sequence of "steps" using "searchers", we want to obtain a state of the graph in which all edges are simultaneously "clear". A search strategy is a sequence of search steps that results in all edges being simultaneously clear. The search number s(G) of a graph G is the smallest number of searchers for which a search strategy exists. A search strategy is monotone if no recontamination ever occurs; it is contiguous if the set of clear edges always forms a connected subgraph; and it is internal if searchers, once placed, can only move along the graph edges (i.e., the removal of searchers and their placement somewhere else is not allowed). Depending on the context, each combination of these characteristics may be desirable. Lapaugh proved that, for any graph G, there exists a monotone search strategy for G using s(G) searchers. Obviously, for any graph G there exists an internal search strategy for G using s(G) searchers, but it is not necessarily monotone. We denote by is(G) (resp. cs(G)) the minimum number of searchers for which there exists a monotone internal (resp. contiguous) search strategy in G. We show that, for any graph G, s(G) = is (G) = cs(G) = 2 s(G). Each of these inequalities can be strict. The last inequality is tight. We actually prove the stronger result stating that, for any graph G, there exists a monotone continguous internal search strategy for G using at most 2 s/G) searchers. As a consequence, the contiguous search number cs is a 2-approximation of pathwidth. Finally, we show that there is a unique obstruction for contiguous search and for monotone internal search in trees, in contrast with standard search which involves exponentially many obstructions, even for trees. We prove this result by giving a complete characterization of those searches in trees.
2022-07-06T15:11:22ZBarrière Figueroa, EulaliaFraigniaud, PierreSantoro, NicolaThilikos, Dimitrios M.In the graph searching problem, we are given a graph whose edges are all "contaminated", and, via a sequence of "steps" using "searchers", we want to obtain a state of the graph in which all edges are simultaneously "clear". A search strategy is a sequence of search steps that results in all edges being simultaneously clear. The search number s(G) of a graph G is the smallest number of searchers for which a search strategy exists. A search strategy is monotone if no recontamination ever occurs; it is contiguous if the set of clear edges always forms a connected subgraph; and it is internal if searchers, once placed, can only move along the graph edges (i.e., the removal of searchers and their placement somewhere else is not allowed). Depending on the context, each combination of these characteristics may be desirable. Lapaugh proved that, for any graph G, there exists a monotone search strategy for G using s(G) searchers. Obviously, for any graph G there exists an internal search strategy for G using s(G) searchers, but it is not necessarily monotone. We denote by is(G) (resp. cs(G)) the minimum number of searchers for which there exists a monotone internal (resp. contiguous) search strategy in G. We show that, for any graph G, s(G) = is (G) = cs(G) = 2 s(G). Each of these inequalities can be strict. The last inequality is tight. We actually prove the stronger result stating that, for any graph G, there exists a monotone continguous internal search strategy for G using at most 2 s/G) searchers. As a consequence, the contiguous search number cs is a 2-approximation of pathwidth. Finally, we show that there is a unique obstruction for contiguous search and for monotone internal search in trees, in contrast with standard search which involves exponentially many obstructions, even for trees. We prove this result by giving a complete characterization of those searches in trees.On the non-uniform complexity of the Graph Isomorphism problem
http://hdl.handle.net/2117/368983
On the non-uniform complexity of the Graph Isomorphism problem
Lozano Boixadors, Antoni; Torán Romero, Jacobo
We study the non-uniform complexity of the Graph Isomorphism (GI) and Graph Automorphism (GA) problems considering the implications of different types of polynomial time reducibilitites from these problems to sparse sets. We show that if GI (or GA) is bounded truth-table or conjunctively reducible to a sparse set, then it is in P; while if we suppose that it is truth-table reducible without restrictions to a sparse set (or, equivalently, that it belongs to P/poly) then the problem is low for MA, the class of sets with publishable proofs. With respect to nondeterministic reductions, contrasting with the fact that GI and GA belong to the class NP¿(co-NP/poly) [Schö 88], we show that if the considered problems are btt strong nondeterministically reducible to a sparse set then they are in co-NP. Some of these results are proved using graph constructions that show new properties of the GI and GA problems.
2022-06-22T10:17:03ZLozano Boixadors, AntoniTorán Romero, JacoboWe study the non-uniform complexity of the Graph Isomorphism (GI) and Graph Automorphism (GA) problems considering the implications of different types of polynomial time reducibilitites from these problems to sparse sets. We show that if GI (or GA) is bounded truth-table or conjunctively reducible to a sparse set, then it is in P; while if we suppose that it is truth-table reducible without restrictions to a sparse set (or, equivalently, that it belongs to P/poly) then the problem is low for MA, the class of sets with publishable proofs. With respect to nondeterministic reductions, contrasting with the fact that GI and GA belong to the class NP¿(co-NP/poly) [Schö 88], we show that if the considered problems are btt strong nondeterministically reducible to a sparse set then they are in co-NP. Some of these results are proved using graph constructions that show new properties of the GI and GA problems.Bounded queries to arbitrary sets
http://hdl.handle.net/2117/329807
Bounded queries to arbitrary sets
Lozano Boixadors, Antoni
We prove that if P superA[k] = P superA[k+1] for some k and an arbitrary set A, then A is reducible to its complement under a relativized nondeterministic conjunctive reduction. This result shows the first known property of arbitrary sets satisfying this condition, and implies some known facts such as Kadin's theorem and its extension to the class C=P.
2020-10-05T13:23:49ZLozano Boixadors, AntoniWe prove that if P superA[k] = P superA[k+1] for some k and an arbitrary set A, then A is reducible to its complement under a relativized nondeterministic conjunctive reduction. This result shows the first known property of arbitrary sets satisfying this condition, and implies some known facts such as Kadin's theorem and its extension to the class C=P.Relativized and positive separations of [delta sub 2 super p] and [fi sub 2 super p]
http://hdl.handle.net/2117/191068
Relativized and positive separations of [delta sub 2 super p] and [fi sub 2 super p]
Lozano Boixadors, Antoni; Torán Romero, Jacobo
2020-06-18T13:29:45ZLozano Boixadors, AntoniTorán Romero, JacoboSelf-reducible sets of small density
http://hdl.handle.net/2117/190992
Self-reducible sets of small density
Lozano Boixadors, Antoni; Torán Romero, Jacobo
We study the complexity of sets that are at the same time self-reducible and sparse or m-reducible to sparse sets. We show that sets of this kind are low for the complexity clases [delta sub 2 super p], [fi sub 2 super p], NP or P, depending on the type of self-reducibility used and on certain restrictions imposed on the query mechanism of the self-reducibility machines. The proof of some of these results is based on graph theoretic properties that hold for the graphs induced by the self-reducibility structures.
2020-06-17T16:17:45ZLozano Boixadors, AntoniTorán Romero, JacoboWe study the complexity of sets that are at the same time self-reducible and sparse or m-reducible to sparse sets. We show that sets of this kind are low for the complexity clases [delta sub 2 super p], [fi sub 2 super p], NP or P, depending on the type of self-reducibility used and on certain restrictions imposed on the query mechanism of the self-reducibility machines. The proof of some of these results is based on graph theoretic properties that hold for the graphs induced by the self-reducibility structures.Complejidad de problemas sobre representaciones sucintas de grafos
http://hdl.handle.net/2117/188934
Complejidad de problemas sobre representaciones sucintas de grafos
Lozano Boixadors, Antoni
2020-05-25T19:37:24ZLozano Boixadors, AntoniTrees whose even-degree vertices induce a path are antimagic
http://hdl.handle.net/2117/133369
Trees whose even-degree vertices induce a path are antimagic
Lozano Boixadors, Antoni; Mora Giné, Mercè; Seara Ojea, Carlos; Tey Carrera, Joaquín
An antimagic labeling of a connected graph G is a bijection from the set of edges E(G) to {1, 2, . . . , |E(G)|} such that all vertex sums are pairwise distinct, where the vertex sum at vertex v is the sum of the labels assigned to edges incident to v. A graph is called antimagic if it has an antimagic labeling. In 1990, Hartsfield and Ringel conjectured that every simple connected graph other than K2 is antimagic; however, the conjecture remains open, even for trees. In this note we prove that trees whose vertices of even degree induce a path are antimagic, extending a result given by Liang, Wong, and Zhu [Discrete Math. 331 (2014) 9–14].
2019-05-23T07:33:59ZLozano Boixadors, AntoniMora Giné, MercèSeara Ojea, CarlosTey Carrera, JoaquínAn antimagic labeling of a connected graph G is a bijection from the set of edges E(G) to {1, 2, . . . , |E(G)|} such that all vertex sums are pairwise distinct, where the vertex sum at vertex v is the sum of the labels assigned to edges incident to v. A graph is called antimagic if it has an antimagic labeling. In 1990, Hartsfield and Ringel conjectured that every simple connected graph other than K2 is antimagic; however, the conjecture remains open, even for trees. In this note we prove that trees whose vertices of even degree induce a path are antimagic, extending a result given by Liang, Wong, and Zhu [Discrete Math. 331 (2014) 9–14].The neighbor-locating-chromatic number of pseudotrees
http://hdl.handle.net/2117/131569
The neighbor-locating-chromatic number of pseudotrees
Hernando Martín, María del Carmen; Mora Giné, Mercè; Pelayo Melero, Ignacio Manuel; Gonzalez, Marisa; Alcón, Liliana
Ak-coloringof a graphGis a partition of the vertices ofGintokindependent sets,which are calledcolors. Ak-coloring isneighbor-locatingif any two vertices belongingto the same color can be distinguished from each other by the colors of their respectiveneighbors. Theneighbor-locating chromatic number¿NL(G) is the minimum cardinalityof a neighbor-locating coloring ofG.In this paper, we determine the neighbor-locating chromatic number of paths, cycles,fans and wheels. Moreover, a procedure to construct a neighbor-locating coloring ofminimum cardinality for these families of graphs is given. We also obtain tight upperbounds on the order of trees and unicyclic graphs in terms of the neighbor-locatingchromatic number. Further partial results for trees are also established.
2019-04-10T04:28:24ZHernando Martín, María del CarmenMora Giné, MercèPelayo Melero, Ignacio ManuelGonzalez, MarisaAlcón, LilianaAk-coloringof a graphGis a partition of the vertices ofGintokindependent sets,which are calledcolors. Ak-coloring isneighbor-locatingif any two vertices belongingto the same color can be distinguished from each other by the colors of their respectiveneighbors. Theneighbor-locating chromatic number¿NL(G) is the minimum cardinalityof a neighbor-locating coloring ofG.In this paper, we determine the neighbor-locating chromatic number of paths, cycles,fans and wheels. Moreover, a procedure to construct a neighbor-locating coloring ofminimum cardinality for these families of graphs is given. We also obtain tight upperbounds on the order of trees and unicyclic graphs in terms of the neighbor-locatingchromatic number. Further partial results for trees are also established.Neighbor-locating colorings in graphs
http://hdl.handle.net/2117/121239
Neighbor-locating colorings in graphs
Hernando Martín, María del Carmen; Mora Giné, Mercè; Pelayo Melero, Ignacio Manuel; Alcón, Liliana; Gutierrez, Marisa
A k -coloring of a graph G is a k -partition ¿ = { S 1 ,...,S k } of V ( G ) into independent sets, called colors . A k -coloring is called neighbor-locating if for every pair of vertices u,v belonging to the same color S i , the set of colors of the neighborhood of u is different from the set of colors of the neighborhood of v . The neighbor-locating chromatic number ¿ NL ( G ) is the minimum cardinality of a neighbor-locating coloring of G . We establish some tight bounds for the neighbor-locating chromatic number of a graph, in terms of its order, maximum degree and independence number. We determine all connected graphs of order n = 5 with neighbor-locating chromatic number n or n - 1. We examine the neighbor-locating chromatic number for two graph operations: join and disjoint union, and also for two graph families: split graphs and Mycielski graphs
2018-09-18T09:56:26ZHernando Martín, María del CarmenMora Giné, MercèPelayo Melero, Ignacio ManuelAlcón, LilianaGutierrez, MarisaA k -coloring of a graph G is a k -partition ¿ = { S 1 ,...,S k } of V ( G ) into independent sets, called colors . A k -coloring is called neighbor-locating if for every pair of vertices u,v belonging to the same color S i , the set of colors of the neighborhood of u is different from the set of colors of the neighborhood of v . The neighbor-locating chromatic number ¿ NL ( G ) is the minimum cardinality of a neighbor-locating coloring of G . We establish some tight bounds for the neighbor-locating chromatic number of a graph, in terms of its order, maximum degree and independence number. We determine all connected graphs of order n = 5 with neighbor-locating chromatic number n or n - 1. We examine the neighbor-locating chromatic number for two graph operations: join and disjoint union, and also for two graph families: split graphs and Mycielski graphsLocating 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 helpful