For a given connected graph G = (V, E), a set D ⊆ V (G) is a doubly connected dominating set if it is dominating and both D and V (G) − D are connected. The cardinality of the minimum doubly connected dominating set in G is the doubly connected domination number. We investigate several properties of doubly connected dominating sets and give some bounds on the doubly connected domination number.
Given an ordered partition $\Pi =\{P_1,P_2, ...,P_t\}$ of the vertex set $V$
of a connected graph $G=(V,E)$, the \emph{partition representation} of a vertex
$v\in V$ with respect to the partition $\Pi$ is the vector
$r(v|\Pi)=(d(v,P_1),d(v,P_2),...,d(v,P_t))$, where $d(v,P_i)$ represents the
distance between the vertex $v$ and the set $P_i$. A partition $\Pi$ of $V$ is
a \emph{resolving partition} of $G$ if different vertices of $G$ have different
partition representations, i.e., for every pair of vertices $u,v\in V$,
$r(u|\Pi)\ne r(v|\Pi)$. The \emph{partition dimension} of $G$ is the minimum
number of sets in any resolving partition of $G$. In this paper we obtain
several tight bounds on the partition dimension of trees
We prove that the domination number γ(T) of a tree T on n ≥ 3 vertices and with n 1 endvertices satisfies inequality γ(T) ≥ n+2−n 1 3 and we characterize the extremal graphs.
The neighbourhood of a vertex v of a graph G is the set N (v) of all verticesThe super domination number of G is the minimum cardinality among all super dominating sets in G. In this article we obtain closed formulas and tight bounds for the super dominating number of lexicographic product graphs in terms of invariants of the factor graphs involved in the product. As a consequence of the study, we show that the problem of finding the super domination number of a graph is NP-Hard.
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