Algorithmic results on variants of domination and domination-related coloring problems in graphs

Abstract

Domination and coloring are two of the most classical and extensively studied graph optimization problems. In recent decades, various fascinating variants of domination and variations integrating concepts of domination and coloring, have been introduced and undergone in-depth exploration. In this thesis, we study the computational complexity of some important variants of domination and domination-related coloring problems. A dominating set of G is a subset S C V, if every vertex not in S is adjacent to at least one vertex in S. For a given graph G, the MINIMUM DOMINATION problem is to compute a dominating set of G with minimum cardinality. We specifically focus on two interesting variations of domination: cosecure domination and semipaired domination, which are defined by imposing some additional conditions. Let G = (V,E) be a graph with no isolated vertices. A dominating set S of G is said to be a cosecure dominating set, if for every vertex v € S there exists a vertex u € V \ S such that v € E and (S\ {v}) U{u} is a dominating set of G. A dominating set S C V of G is said to be a semipaired dominating set, if S can be partitioned into cardinality 2 subsets such that the vertices in each of these subsets are at distance at most two from each other. A coloring (or proper coloring) of G is an assignment of colors to the vertices of G such that if two vertices are adjacent, then they must be assigned different colors. For a given graph G, the MINIMUM COLORING problem is to find a coloring of G using minimum number of colors. We explore two intriguing variations of domination-related coloring: total dominator coloring and domination coloring, defined introducing supplementary conditions of domination in coloring. For a graph G without any isolated vertex, a coloring of G is called a total dominator coloring, if each vertex dominates some color class other than its own. A coloring of G is termed as a domination coloring, if each vertex dominates some color class and each color class is dominated by some vertex. All the graph optimization problems, mentioned above, are known to be NP-hard for general graphs. To address this challenge, one strategy is to explore the problem within the context of restricted graph classes, as real-world problems often result in graphs with distinctive properties. In this thesis, we adopt this strategy and examine the computational complexity of these problems across various subclasses of graphs, distinguished by their structural properties. Our analysis reveals specific graph classes where these problems remain NP-hard and we develop efficient algorithms for solving these problems on certain special graph classes. Additionally, we work on approximation aspects of the problems by proving some results regarding bounds on approximation ratio, presenting some approximation algorithms, and demonstrating approximation hardness for selected instances of these problems.