A study on partial domination in graphs

A study on partial domination in graphs

Mathematics

The theory of domination is one of the most studied fields in graph theory. Many new domination parameters have been defined and studied so far. One such parameter that was introduced in 2017 is partial domination number. For a graph G = (V,E) and for a p ∈ (0,1], a subset S of V(G) is said to partially dominate or p-dominate G if |N[S]| ≥ p|V(G)|. The cardinality of a smallest p-dominating set is called the p-domination number and it is denoted by γp(G). In scenarios wherein domination concepts are applied, partial domination concepts can also be applied with the added advantage of being able to dominate the underlying graph partially, when the need arises. This advantage makes this parameter appear unique amongst most other domination parameters. We present some basic properties of partial dominating sets, some properties related to particular values of p, some properties related to the eccentricity of a p-dominating set, some results in the line of classical domination and characterization of minimal and minimum p-dominating sets. Then we study partial domination in the context of prisms of graphs. We give some bounds for partial domination numbers of prisms of graphs G in terms of partial domination numbers of G for particular values of p. We define universal γp-fixers and universal γp-doublers and we characterize paths, cycles and complete bipartite graphs which are universal γ 1 2 - fixers and universal γ 1 2 - doublers. Then we concentrate on establishing a domination chain in the context of partial
domination, which we call as ‘partial domination chain’.

Nithya, L Philo - 1740088

CHRIST (Deemed to be University)

English

PhD