On the metric dimension of corona product graphs

Given a set of vertices $S=\{v_1,v_2,...,v_k\}$ of a connected graph $G$, the metric representation of a vertex $v$ of $G$ with respect to $S$ is the vector $r(v|S)=(d(v,v_1),d(v,v_2),...,d(v,v_k))$, where $d(v,v_i)$, $i\in \{1,...,k\}$ denotes the distance between $v$ and $v_i$. $S$ is a resolving set for $G$ if for every pair of vertices $u,v$ of $G$, $r(u|S)\ne r(v|S)$. The metric dimension of $G$, $dim(G)$, is the minimum cardinality of any resolving set for $G$. Let $G$ and $H$ be two graphs of order $n_1$ and $n_2$, respectively. The corona product $G\odot H$ is defined as the graph obtained from $G$ and $H$ by taking one copy of $G$ and $n_1$ copies of $H$ and joining by an edge each vertex from the $i^{th}$-copy of $H$ with the $i^{th}$-vertex of $G$. For any integer $k\ge 2$, we define the graph $G\odot^k H$ recursively from $G\odot H$ as $G\odot^k H=(G\odot^{k-1} H)\odot H$. We give several results on the metric dimension of $G\odot^k H$. For instance, we show that given two connected graphs $G$ and $H$ of order $n_1\ge 2$ and $n_2\ge 2$, respectively, if the diameter of $H$ is at most two, then $dim(G\odot^k H)=n_1(n_2+1)^{k-1}dim(H)$. Moreover, if $n_2\ge 7$ and the diameter of $H$ is greater than five or $H$ is a cycle graph, then $dim(G\odot^k H)=n_1(n_2+1)^{k-1}dim(K_1\odot H).

Boundary powerful k-alliances in graphs

A global boundary defensive k-alliance in a graph G = (V, E) is a dominating set S of vertices of G with the property that every vertex in 5 has fc more neighbors in 5 than it has outside of S. A global boundary offensive fc-alliance in a graph G is a set S of vertices of G with the property that every vertex in V - S has fc more neighbors in S than it has outside of S. We define a global boundary powerful fc-alliance as a set 5 of vertices of G, which is both global boundary defensive kalliance and global boundary offensive (k + 2)-alliance. In this paper we study mathematical properties of boundary powerful k-alliances. In particular, we obtain several bounds (closed formulas for the case of regular graphs) on the cardinality of every global boundary powerful k-alliance. In addition, we consider the case in which the vertex set of a graph G can be partitioned' into two boundary powerful fcalliances, showing that, in such a case, k = - 1 and, if G is amp;d-regular, its algebraic connectivity is equal to amp;d +1

Alliance free sets in Cartesian product graphs

10.1016/j.dam.2013.01.01