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：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有 $j$ 个邻居，则称 $D$ 为 $G$ 的 $[1,j]$-支配集。最小 $[1,j]$-支配问题即寻找此类最小集合 $D$ 的问题。给定正整数 $k$ 和图 $G = (V, E)$，$[1,j]$-支配判定问题旨在判断 $G$ 是否存在基数不超过 $k$ 的 $[1,j]$-支配集。对于常数 $j$，在分裂图中已获得多项式时间算法，这与经典支配集问题在分裂图中是 NP 难问题形成对比。该结果促使我们研究限制条件 $j$ 对 $[1,j]$-支配问题在不同图类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值等于 $[1,j]$-支配的最小值，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，提出了一种在非真区间图上计算最小 $[1,2]$-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。