A subset $S$ of vertices in a graph $G=(V, E)$ is Dominating Set if each vertex in $V(G)\setminus S$ is adjacent to at least one vertex in $S$. Chellali et al. in 2013, by restricting the number of neighbors in $S$ of a vertex outside $S$, introduced the concept of $[1,j]$-dominating set. A set $D \subseteq V$ of a graph $G = (V, E)$ is called $[1,j]$-Dominating Set of $G$ if every vertex not in $D$ has at least one neighbor and at most $j$ neighbors in $D$. The Minimum $[1,j]$-Domination problem is the problem of finding the minimum set $D$. Given a positive integer $k$ and a graph $G = (V, E)$, the $[1,j]$-Domination Decision problem is to decide whether $G$ has $[1,j]$-dominating set of cardinality at most $k$. A polynomial-time algorithm was obtained in split graphs for a constant $j$ in contrast to the classic Dominating Set problem which is NP-hard in split graphs. This result motivates us to investigate the effect of restriction $j$ on the complexity of $[1,j]$-domination problem on various classes of graphs. Although for $j\geq 3$, it has been proved that the minimum of classical domination is equal to minimum $[1,j]$-domination in interval graphs, the complexity of finding the minimum $[1,2]$-domination in interval graphs is still outstanding. In this paper, we propose a polynomial-time algorithm for computing a minimum $[1,2]$ on non-proper interval graphs by a dynamic programming technique. Next, on the negative side, we show that the minimum $[1,2]$-dominating set problem on circle graphs is $NP$-complete.

翻译：图$G=(V, E)$中的顶点子集$S$称为支配集，如果$V(G)\setminus S$中的每个顶点都与$S$中至少一个顶点相邻。Chellali等人于2013年通过限制$S$外部顶点在$S$中的邻居数量，引入了$[1,j]$-支配集的概念。图$G = (V, E)$中的集合$D \subseteq V$称为$G$的$[1,j]$-支配集，如果不在$D$中的每个顶点在$D$中至少有一个邻居且至多$j$个邻居。最小$[1,j]$-支配问题是指寻找最小集合$D$的问题。给定正整数$k$和图$G = (V, E)$，$[1,j]$-支配判定问题是判断$G$是否具有基数不超过$k$的$[1,j]$-支配集。在分裂图中，对于常数$j$，该问题存在多项式时间算法，而经典支配集问题在分裂图中是NP难的。这一结果促使我们研究限制$j$对各类图上$[1,j]$-支配问题复杂性的影响。尽管对于$j\geq 3$，已证明在区间图中经典支配的最小值等于最小$[1,j]$-支配，但寻找区间图中最小$[1,2]$-支配的复杂度问题仍未解决。本文通过动态规划技术，提出了一种计算非真区间图中最小$[1,2]$-支配的多项式时间算法。此外，在负面结果方面，我们证明圆形图上的最小$[1,2]$-支配集问题是NP完全的。