A subset $S$ of vertices in a graph $G=(V, E)$ is a 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 a $[1,j]$-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 $[1,j]$-dominating 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 a $[1,j]$-dominating set of cardinality at most $k$. A polynomial-time algorithm was obtained in split graphs for a constant $j$ in contrast to the Dominating Set problem which is NP-hard for 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 $[1,2]$-dominating set on 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：在图$G=(V, E)$中，若顶点子集$S$满足$V(G)\setminus S$中的每个顶点都与$S$中至少一个顶点相邻，则称$S$为支配集。Chellali等人于2013年通过限制$S$外顶点在$S$中的邻居数量，提出了$[1,j]$-支配集的概念。对于图$G = (V, E)$，若子集$D \subseteq V$满足不在$D$中的每个顶点在$D$中至少有一个且至多$j$个邻居，则称$D$为$G$的$[1,j]$-支配集。最小$[1,j]$-支配问题旨在寻找基数最小的$[1,j]$-支配集$D$。给定正整数$k$和图$G = (V, E)$，$[1,j]$-支配判定问题需要判断$G$是否存在基数不超过$k$的$[1,j]$-支配集。针对常数$j$，已有研究在分裂图中提出了多项式时间算法，而经典支配集问题在分裂图中是NP困难的。这一结果促使我们探究限制条件$j$对$[1,j]$-支配问题在不同图类中计算复杂度的影响。尽管已有证明表明，当$j\geq 3$时，区间图上经典支配的最小基数等于$[1,j]$-支配的最小基数，但求解区间图最小$[1,2]$-支配集的复杂度问题仍未解决。本文通过动态规划技术，提出了计算区间图最小$[1,2]$-支配集的多项式时间算法。另一方面，我们证明了圆图上的最小$[1,2]$-支配集问题是NP完全的。