Approximation hardness of domination problems on generalized convex graphs

Abstract

The domination problem and its variants in bipartite graphs are computationally challenging, known to be $\mathrm{NP}$-complete and hard to approximate. However, for convex bipartite graphs, it becomes polynomial-time solvable, raising questions about the boundaries of tractability and intractability, as well as approximability and inapproximability, within bipartite graph subclasses. This study examines the approximation hardness of the domination problem for generalized convex graphs, a subclass of bipartite graphs that extends convex bipartite graphs. We explore the approximation hardness for various domination problem variants, including total, connected, paired, and independent domination. Previous research has highlighted the critical role of the $(t,\Delta )$-tree convex graph parameters in determining computational complexity, demonstrating polynomial-time solvability when both t and Δ are bounded. Extending these findings, our research establishes that unbounded t or Δ results in $\mathrm{APX}$-hardness for all examined domination variants. Notably, this result encompasses star and comb convex bipartite graphs.