Partial domination of network modelling

Abstract

<abstract><p>Partial domination was proposed in 2017 on the basis of domination theory, which has good practical application background in communication network. Let $ G = (V, E) $ be a graph and $ \mathcal{F} $ be a family of graphs, a subset $ S\subseteq V $ is called an $ \mathcal{F} $-isolating set of $ G $ if $ G[V\backslash N_G[S]] $ does not contain $ F $ as a subgraph for all $ F\in\mathcal{F} $. The subset $ S $ is called an isolating set of $ G $ if $ \mathcal{F} = \{K_2\} $ and $ G[V\backslash N_G[S]] $ does not contain $ K_2 $ as a subgraph. The isolation number of $ G $ is the minimum cardinality of an isolating set of $ G $, denoted by $ \iota(G) $. The hypercube network and $ n $-star network are the basic models for network systems, and many more complex network structures can be built from them. In this study, we obtain the sharp bounds of the isolation numbers of the hypercube network and $ n $-star network.</p></abstract>