A Survey on Variant Domination Problems in Geometric Intersection Graphs

Abstract

Given a graph [Formula: see text] with vertex set [Formula: see text], a subset [Formula: see text] of [Formula: see text] is called a dominating set of [Formula: see text] if every vertex is either in [Formula: see text] or is adjacent to a vertex in [Formula: see text]. There are a lot of variants for dominating sets, such as connected dominating sets, total dominating sets, total restricted dominating sets, secure (connected, total) dominating sets, etc. Geometric intersection graphs are graphs for which there is a bijection [Formula: see text] between the vertices and a set [Formula: see text] of geometric objects (for example, disks, rectangles, etc.) such that there is an edge between two vertices [Formula: see text] and [Formula: see text] if and only if the objects [Formula: see text] and [Formula: see text] intersect. In this paper, we offer a survey about complexity and algorithmic results on variant domination problems in geometric intersection graphs.