On $[1,2]$-Domination in Interval and Circle Graphs

Abstract

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.

Figures (10)

• Figure 1: Illustrating (a) an interval graph $G$ of order $10$, (b) a set of intervals corresponding to the vertex set of $G$
• Figure 2: (a) The circle graph $G$ on $8$ vertices (b)The circle representation of $G$ (c)The interval representation of $G$
• Figure 3: Inclusion relations among well-studied classes of graphs tripathi2022complexity - Complexity of $[1,j]$-dominating set restricted to various classes of graphs in a hierarchy of graphs - $NPc$ indicate that the problem is $NP$-complete, $P$ indicates the problem has a polynomial algorithm, and the question mark indicates the open problem.
• Figure 4: (a) A set of intervals corresponding to the vertex set of the interval graph $G$; (b) a numbering $(1,2,\ldots , 10)$ of the vertices of $G$ satisfying the condition of Corollary \ref{['theo:RR']}; and (c) values $\mathtt{low}(i)$ and $\mathtt{maxlow}(i)$ for all $i\in [ 1, 10 ]$
• Figure 5: Illustrating an $[1,2]$-dominating set $D$ of $G[ 1, i]$ such that $a\notin D$ for all $a\in [i',i]$; (a) no vertex of $[ \mathtt{low} ( j+1,i), j)$ is in $D$ and (b) exactly one vertex $k\in [ \mathtt{low} ( j+1,i), j)$ is in $D$. Note that a vertex with label 1 is in $D$ and a vertex with label 0 is not in $D$.

Theorems & Definitions (25)

• Lemma 1
• Lemma 2: ramalingam1988unified, Theorem 2.1
• Lemma 3: poureidi2022algorithm
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof
• Lemma 7