Lower Bounds on the Distance Domination Number of a Graph
2017
For an integer $k \ge 1$, a (distance) $k$-dominating set of a connected graph $G$ is a set $S$ of vertices of $G$ such that every vertex of $V(G) \setminus S$ is at distance at most~$k$ from some vertex of $S$. The $k$-domination number, $\gamma_k(G)$, of $G$ is the minimum cardinality of a $k$-dominating set of $G$. In this paper, we establish lower bounds on the $k$-domination number of a graph in terms of its diameter, radius, and girth. We prove that for connected graphs $G$ and $H$, $\gamma_k(G \times H) \ge \gamma_k(G) + \gamma_k(H) -1$, where $G \times H$ denotes the direct product of $G$ and $H$.
Keywords:
- Correction
- Source
- Cite
- Save
- Machine Reading By IdeaReader
0
References
6
Citations
NaN
KQI