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 $$d\ge 3$$.