21 research outputs found
The Simultaneous Strong Resolving Graph and the Simultaneous Strong Metric Dimension of Graph Families
We consider in this work a new approach to study the simultaneous strong metric dimension of graphs families, while introducing the simultaneous version of the strong resolving graph. In concordance, we consider here connected graphs G whose vertex sets are represented as V(G), and the following terminology. Two vertices u,v is an element of V(G) are strongly resolved by a vertex w is an element of V(G), if there is a shortest w-v path containing u or a shortest w-u containing v. A set A of vertices of the graph G is said to be a strong metric generator for G if every two vertices of G are strongly resolved by some vertex of A. The smallest possible cardinality of any strong metric generator (SSMG) for the graph G is taken as the strong metric dimension of the graph G. Given a family F of graphs defined over a common vertex set V, a set S subset of V is an SSMG for F, if such set S is a strong metric generator for every graph G is an element of F. The simultaneous strong metric dimension of F is the minimum cardinality of any strong metric generator for F, and is denoted by Sds(F). The notion of simultaneous strong resolving graph of a graph family F is introduced in this work, and its usefulness in the study of Sds(F) is described. That is, it is proved that computing Sds(F) is equivalent to computing the vertex cover number of the simultaneous strong resolving graph of F. Several consequences (computational and combinatorial) of such relationship are then deduced. Among them, we remark for instance that we have proved the NP-hardness of computing the simultaneous strong metric dimension of families of paths, which is an improvement (with respect to the increasing difficulty of the problem) on the results known from the literature
Defensive alliances in graphs: a survey
A set of vertices of a graph is a defensive -alliance in if
every vertex of has at least more neighbors inside of than outside.
This is primarily an expository article surveying the principal known results
on defensive alliances in graph. Its seven sections are: Introduction,
Computational complexity and realizability, Defensive -alliance number,
Boundary defensive -alliances, Defensive alliances in Cartesian product
graphs, Partitioning a graph into defensive -alliances, and Defensive
-alliance free sets.Comment: 25 page
Further new results on strong resolving partitions for graphs
A set W of vertices of a connected graph G strongly resolves two different vertices x, y is not an element of W if either d(G) (x, W) = d(G) (x, y) + d(G) (y, W) or d(G) (y, W) = d(G )(y, x) + d(G) (x, W), where d(G) (x, W) = min{d(x,w): w is an element of W} and d (x,w) represents the length of a shortest x - w path. An ordered vertex partition Pi = {U-1, U-2,...,U-k} of a graph G is a strong resolving partition for G, if every two different vertices of G belonging to the same set of the partition are strongly resolved by some other set of Pi. The minimum cardinality of any strong resolving partition for G is the strong partition dimension of G. In this article, we obtain several bounds and closed formulae for the strong partition dimension of some families of graphs and give some realization results relating the strong partition dimension, the strong metric dimension and the order of graphs
On the Packing Partitioning Problem on Directed Graphs
This work is aimed to continue studying the packing sets of digraphs via the perspective of
partitioning the vertex set of a digraph into packing sets (which can be interpreted as a type of vertex
coloring of digraphs) and focused on finding the minimum cardinality among all packing partitions
for a given digraph D, called the packing partition number of D. Some lower and upper bounds on
this parameter are proven, and their exact values for directed trees are given in this paper. In the case
of directed trees, the proof results in a polynomial-time algorithm for finding a packing partition of
minimum cardinality. We also consider this parameter in digraph products. In particular, a complete
solution to this case is presented when dealing with the rooted products
Error-Correcting codes fromk-resolving sets
We demonstrate a construction of error-correcting codes from graphs by
means of k-resolving sets, and present a decoding algorithm which makes use
of covering designs. Along the way, we determine the k-metric dimension of
grid graphs (i.e., Cartesian products of paths)
Coloring of two-step graphs: open packing partitioning of graphs
The two-step graphs are revisited by studying their chromatic numbers in this
paper. We observe that the problem of coloring of two-step graphs is equivalent
to the problem of vertex partitioning of graphs into open packing sets. With
this remark in mind, it can be considered as the open version of the well-known
-distance coloring problem as well as the dual version of total domatic
problem.
The minimum for which the two-step graph of a graph
admits a proper coloring assigning colors to the vertices is called the
open packing partition number of , that is,
p_{o}(G)=\chi\big{(}\mathcal{N}(G)\big{)}. We give some sharp lower and upper
bounds on this parameter as well as its exact value when dealing with some
families of graphs like trees. Relations between and some well-know
graph parameters have been investigated in this paper. We study this vertex
partitioning in the Cartesian, direct and lexicographic products of graphs. In
particular, we give an exact formula in the case of lexicographic product of
any two graphs. The NP-hardness of the problem of computing this parameter is
derived from the mentioned formula. Graphs for which equals the
clique number of are also investigated
Further Contributions on the Outer Multiset Dimension of Graphs
The outer multiset dimension dim ms(G) of a graph G is the cardinality of a smallest set of vertices that uniquely recognize all the vertices outside this set by using multisets of distances to the set. It is proved that dim ms(G) = n(G) - 1 if and only if G is a regular graph with diameter at most 2. Graphs G with dim ms(G) = 2 are described and recognized in polynomial time. A lower bound on the lexicographic product of G and H is proved when H is complete or edgeless, and the extremal graphs are determined. It is proved that dimms(Ps□Pt)=3 for s≥ t≥ 2.15 página
A constructive characterization of vertex cover Roman trees
A Roman dominating function on a graph G = (V (G), E (G)) is a function f : V (G) -> {0, 1, 2} satisfying the condition that every vertex u for which f (u) = 0 is adjacent to at least one vertex v for which f (v) = 2. The Roman dominating function f is an outer-independent Roman dominating function on G if the set of vertices labeled with zero under f is an independent set. The outer-independent Roman domination number gamma(oiR) (G) is the minimum weight w(f ) = Sigma(v is an element of V), ((G)) f(v) of any outer-independent Roman dominating function f of G. A vertex cover of a graph G is a set of vertices that covers all the edges of G. The minimum cardinality of a vertex cover is denoted by alpha(G). A graph G is a vertex cover Roman graph if gamma(oiR) (G) = 2 alpha(G). A constructive characterization of the vertex cover Roman trees is given in this article
Roman domination in direct product graphs and rooted product graphs1
Let G be a graph with vertex set V(G). A function f : V(G) -> {0, 1, 2) is a Roman dominating function on G if every vertex v is an element of V(G) for which f(v) = 0 is adjacent to at least one vertex u is an element of V(G) such that f(u) = 2. The Roman domination number of G is the minimum weight omega(f) = Sigma(x is an element of V(G)) f(x) among all Roman dominating functions f on G. In this article we study the Roman domination number of direct product graphs and rooted product graphs. Specifically, we give several tight lower and upper bounds for the Roman domination number of direct product graphs involving some parameters of the factors, which include the domination, (total) Roman domination, and packing numbers among others. On the other hand, we prove that the Roman domination number of rooted product graphs can attain only three possible values, which depend on the order, the domination number, and the Roman domination number of the factors in the product. In addition, theoretical characterizations of the classes of rooted product graphs achieving each of these three possible values are given.The second author (Iztok Peterin) has been partially supported by the Slovenian Research Agency by the projects No. J1-1693 and J1-9109. The last author (Ismael G. Yero) has been partially supported by "Junta de Andalucia", FEDER-UPO Research and Development Call, reference number UPO1263769
Global defensive k-alliances in directed graphs: combinatorial and computational issues
In this paper we define the global defensive k-alliance (number) in a digraph D, and give several bounds on this parameter with characterizations of all digraphs attaining the bounds. In particular, for the case k = -1, we give a lower (an upper) bound on this parameter for directed trees (rooted trees). Moreover, the characterization of all directed trees (rooted trees) for which the equality holds is given. Finally, we show that the problem of finding the global defensive k-alliance number of a digraph is NP-hard for any suitable non-negative value of k, and in contrast with it, we also show that finding a minimum global defensive (-1)-alliance for any rooted tree is polynomial-time solvable