Abstract
A subset $S$ of the vertices of $G = (V, E)$ is an $[a, b]$-set if for every vertex $v$ not in $S$ we have the number of neighbors of $v$ in $S$ is between $a$ and $b$ for non-negative integers $a$ and $b$, that is, every vertex $v$ not in $S$ is adjacent to at least $a$ but not more than $b$ vertices in $S$. The minimum cardinality of an $[a, b]$-set of $G$ is called the $[a, b]$-domination number of $G$. The $[a, b]$-domination problem is to determine the $[a, b]$-domination number of a graph. In this paper, we show that the [2,b]-domination problem is NP-complete for $b$ at least $3$, and the [1,2]-total domination problem is NP-complete. We also determine the [1,2]-total domination and [1,2] domination numbers of toroidal grids with three rows and four rows.
Highlights
Let G be a graph, S ⊆ V(G), v ∈ V(G)
We show that the [2,b]-domination problem is NP-complete for b ≥ 3, and the [1,2]-total domination problem is NP-complete
We will prove the following result: Theorem 1 The [2,b]-domination problem is NP-complete for b ≥ 3
Summary
Let G be a graph, S ⊆ V(G), v ∈ V(G). The open neighborhood of v in S , {u|uv ∈ E(G), u ∈ S }, is denoted by NS (v).We write NS [v] = {v} ∪ NS (v). [a, b]-total domination) number of a graph.
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