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 $V(G)\setminus S$ 中的每个顶点均与 $S$ 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有 $j$ 个邻居，则称 $D$ 为 $G$ 的 $[1,j]$-支配集。最小 $[1,j]$-支配集问题即寻找最小规模的 $[1,j]$-支配集 $D$。给定正整数 $k$ 和图 $G = (V, E)$，$[1,j]$-支配判定问题旨在判断 $G$ 是否存在规模不超过 $k$ 的 $[1,j]$-支配集。对于固定常数 $j$，已在分裂图中获得多项式时间算法，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条件 $j$ 对 $[1,j]$-支配问题在各类图上计算复杂度的影响。尽管已证明当 $j\geq 3$ 时，区间图上经典支配的最小规模等于最小 $[1,j]$-支配规模，但寻找区间图上最小 $[1,2]$-支配集的复杂度仍未解决。本文通过动态规划技术，提出了在区间图上计算最小 $[1,2]$-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。