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 $S$ of vertices of a graph $G$ is a defensive $k$-alliance in $G$ if every vertex of $S$ has at least $k$ more neighbors inside of $S$ 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 $k$-alliance number, Boundary defensive $k$-alliances, Defensive alliances in Cartesian product graphs, Partitioning a graph into defensive $k$-alliances, and Defensive $k$-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

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

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

Further results on packing related parameters in graphs

Given a graph G = (V, E), a set B subset of V (G) is a packing in G if the closed neighborhoods of every pair of distinct vertices in B are pairwise disjoint. The packing number rho(G) of G is the maximum cardinality of a packing in G. Similarly, open packing sets and open packing number are defined for a graph G by using open neighborhoods instead of closed ones. We give several results concerning the (open) packing number of graphs in this paper. For instance, several bounds on these packing parameters along with some Nordhaus-Gaddum inequalities are given. We characterize all graphs with equal packing and independence numbers and give the characterization of all graphs for which the packing number is equal to the independence number minus one. In addition, due to the close connection between the open packing and total domination numbers, we prove a new upper bound on the total domination number gamma(t)(T) for a tree T of order n >= 2 improving the upper bound gamma(t)(T) <= (n + s)/2 given by Chellali and Haynes in 2004, in which s is the number of support vertices of T

Independent transversal total domination versus total domination in trees

A subset of vertices in a graph G is a total dominating set if every vertex in G is adjacent to at least one vertex in this subset. The total domination number of G is the minimum cardinality of any total dominating set in G and is denoted by gamma(t)(G). A total dominating set of G having nonempty intersection with all the independent sets of maximum cardinality in G is an independent transversal total dominating set. The minimum cardinality of any independent transversal total dominating set is denoted by gamma(u) (G). Based on the fact that for any tree T, gamma(t) (T) <= gamma(u) (T) <= gamma(t) (T) + 1, in this work we give several relationship(s) between gamma(u) (T) and gamma(t) (T) for trees T which are leading to classify the trees which are satisfying the equality in these bound

Relating the Outer-Independent Total Roman Domination Number with Some Classical Parameters of Graphs

For a given graph G without isolated vertex we consider a function f : V (G) -> {0,1, 2}. For every i is an element of {0,1, 2}, let V-i = {v is an element of V (G) : f (v) = i}. The function f is known to be an outer-independent total Roman dominating function for the graph G if it is satisfied that; (i) every vertex in V-0 is adjacent to at least one vertex in V-2; (ii) V-0 is an independent set; and (iii) the subgraph induced by V-1 boolean OR V-2 has no isolated vertex. The minimum possible weight omega(f) = Sigma(v is an element of V(G)) f(v) among all outer-independent total Roman dominating functions for G is called the outer-independent total Roman domination number of G. In this article we obtain new tight bounds for this parameter that improve some well-known results. Such bounds can also be seen as relationships between this parameter and several other classical parameters in graph theory like the domination, total domination, Roman domination, independence, and vertex cover numbers. In addition, we compute the outer-independent total Roman domination number of Sierpinski graphs, circulant graphs, and the Cartesian and direct products of complete graphs