# Convex and isometric domination of (weak) dominating pair graphs

## TL;DR

This paper investigates the computational complexity of convex and isometric domination problems in special classes of graphs, providing efficient algorithms for some classes and NP-completeness results for others.

## Contribution

It introduces algorithms for finding minimum convex and isometric dominating sets in weak and chordal dominating pair graphs, and proves NP-completeness for certain subclasses.

## Key findings

- Efficient algorithm for minimum isometric dominating set in weak dominating pair graphs.
- NP-completeness of convex dominating set decision problem in chordal weak dominating pair graphs.
- Polynomial-time algorithm for minimum convex dominating set in chordal dominating pair graphs.

## Abstract

A set $D$ of vertices in a graph $G$ is a dominating set if every vertex of $G$, which is not in $D$, has a neighbor in $D$. A set of vertices $D$ in $G$ is convex (respectively, isometric), if all vertices in all shortest paths (respectively, all vertices in one of the shortest paths) between any two vertices in $D$ lie in $D$. The problem of finding a minimum convex dominating (respectively, isometric dominating) set is considered in this paper from algorithmic point of view. For the class of weak dominating pair graphs (i.e.,~the graphs that contain a dominating pair, which is a pair of vertices $x,y\in V(G)$ such that vertices of any path between $x$ and $y$ form a dominating set), we present an efficient algorithm that finds a minimum isometric dominating set of such a graph. On the other hand, we prove that even if one restricts to weak dominating pair graphs that are also chordal graphs, the problem of deciding whether there exists a convex dominating set bounded by a given arbitrary positive integer is NP-complete. By further restricting the class of graphs to chordal dominating pair graphs (i.e.,~the chordal graphs in which every connected induced subgraph has a dominating pair) we are able to find a polynomial time algorithm that determines the minimum size of a convex dominating set of such a graph.