Distance- $$d$$ independent set problems for bipartite and chordal graphs

The paper studies a generalization of the Independent Set problem (IS for short). A distance- $$d$$ independent set for an integer $$d\ge 2$$ in an unweighted graph $$G = (V, E)$$ is a subset $$S\subseteq V$$ of vertices such that for any pair of vertices $$u, v \in S$$ , the distance between $$u$$ and $$v$$ is at least $$d$$ in $$G$$ . Given an unweighted graph $$G$$ and a positive integer $$k$$ , the Distance- $$d$$ Independent Set problem (D $$d$$ IS for short) is to decide whether $$G$$ contains a distance- $$d$$ independent set $$S$$ such that $$|S| \ge k$$ . D2IS is identical to the original IS. Thus D2IS is $$\mathcal{NP}$$ -complete even for planar graphs, but it is in $$\mathcal{P}$$ for bipartite graphs and chordal graphs. In this paper we investigate the computational complexity of D $$d$$ IS, its maximization version MaxD $$d$$ IS, and its parameterized version ParaD $$d$$ IS( $$k$$ ), where the parameter is the size of the distance- $$d$$ independent set: (1) We first prove that for any $$\varepsilon >0$$ and any fixed integer $$d\ge 3$$ , it is $$\mathcal{NP}$$ -hard to approximate MaxD $$d$$ IS to within a factor of $$n^{1/2-\varepsilon }$$ for bipartite graphs of $$n$$ vertices, and for any fixed integer $$d\ge 3$$ , ParaD $$d$$ IS( $$k$$ ) is $$\mathcal{W}[1]$$ -hard for bipartite graphs. Then, (2) we prove that for every fixed integer $$d\ge 3$$ , D $$d$$ IS remains $$\mathcal{NP}$$ -complete even for planar bipartite graphs of maximum degree three. Furthermore, (3) we show that if the input graph is restricted to chordal graphs, then D $$d$$ IS can be solved in polynomial time for any even $$d\ge 2$$ , whereas D $$d$$ IS is $$\mathcal{NP}$$ -complete for any odd $$d\ge 3$$ . Also, we show the hardness of approximation of MaxD $$d$$ IS and the $$\mathcal{W}[1]$$ -hardness of ParaD $$d$$ IS( $$k$$ ) on chordal graphs for any odd $$d\ge 3$$ .