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For each vertex v in a graph G, let there be associated a subgraph $H_v$ of G. The vertex v is said to dominate $H_v$ as well as dominate each vertex and edge of $H_v$. A set S of vertices of G is called a full dominating set if every vertex of G is dominated by some vertex of S, as is every edge of G. The minimum cardinality of a full dominating set of G is its full domination number ${\gamma }_{FH}\left(G\right)$. A full dominating set of G of cardinality ${\gamma }_{FH}\left(G\right)$ is called a ${\gamma }_{FH}$-set of G. We study three types of full domination in graphs: full...

For an ordered set $W=\{w_1,w_2,\cdots ,w_k\}$ of vertices and a vertex $v$ in a connected graph $G$, the (metric) representation of $v$ with respect to $W$ is the $k$-vector $r\left(v\right|W)=(d(v,w_1),d(v,w_2),\cdots ,d(v,w_k))$, where $d(x,y)$ represents the distance between the vertices $x$ and $y$. The set $W$ is a resolving set for $G$ if distinct vertices of $G$ have distinct representations with respect to $W$. A resolving set of minimum cardinality is called a minimum resolving set or a basis and the cardinality of a basis for $G$ is its dimension $dimG$. A set $S$ of vertices in $G$ is a dominating set...