Red Domination in Perfect Elimination Bipartite Graphs

TL;DR

This paper investigates the computational complexity of the red domination problem in various bipartite graph classes, establishing NP-completeness results and providing efficient algorithms for specific subclasses.

Contribution

It proves NP-completeness for perfect elimination bipartite graphs and introduces a linear delay enumeration algorithm for convex bipartite graphs.

Findings

NP-complete for perfect elimination bipartite graphs

Linear time calculation of the number of dominating sets in convex bipartite graphs

Linear delay enumeration algorithm for dominating sets

Abstract

The $k$ red domination problem for a bipartite graph $G=(X,Y,E)$ is to find a subset $D \subseteq X$ of cardinality at most $k$ that dominates vertices of $Y$. The decision version of this problem is NP-complete for general bipartite graphs but solvable in polynomial time for chordal bipartite graphs. We strengthen that result by showing that it is NP-complete for perfect elimination bipartite graphs. We present a tight upper bound on the number of such sets in bipartite graphs, and show that we can calculate that number in linear time for convex bipartite graphs. We present a linear space linear delay enumeration algorithm that needs only linear preprocessing time.