Approximating Domination on Intersection Graphs of Paths on a Grid

Abstract

A graph G is called ${B}_k$-EPG (resp., ${B}_k$-VPG), for some constant $k\ge 0$, if it has a string representation on an axis-parallel grid such that each vertex is a path with at most k bends and two vertices are adjacent in G if and only if the corresponding strings share at least one grid edge (resp., the corresponding strings intersect each other). If two adjacent strings of a B$_k$-VPG graph intersect each other exactly once, then the graph is called a one-string B$_k$-VPG graph.

In this paper, we study the Minimum Dominating Set problem on $\textsc {B}_1\text {-}{\textsc {EPG}}$ and $\textsc {B}_1\text {-}{\textsc {VPG}}$ graphs. We first give an O(1)-approximation algorithm on one-string $\textsc {B}_1\text {-}{\textsc {VPG}}$ graphs, providing the first constant-factor approximation algorithm for this problem. Moreover, we show that the Minimum Dominating Set problem is APX-hard on $\textsc {B}_1\text {-}{\textsc {EPG}}$ graphs, ruling out the possibility of a PTAS unless P = NP. Finally, to complement our APX-hardness result, we give constant-factor approximation algorithms for the Minimum Dominating Set problem on two non-trivial subclasses of $\textsc {B}_1\text {-}{\textsc {EPG}}$ graphs.