An Algorithm for the Domination Number and Neighbor-component Order Connectivity of a Unicycle

The graph parameter domination is considered perhaps one of the fastest-growing areas in graph theory since the fundamental efforts of Berge and Ore in 1962. The domination number of a graph may be determined using the parameter neighbor-component order connectivity which is defined as the minimum number of nodes such that removing these nodes and their adjacent neighbors ensure all remaining components have order less than some given threshold value. Given a threshold value of one, the neighbor-component order connectivity of a graph is equivalent to the well-known parameter domination number of the graph. The problem of computing the domination number of an arbitrary graph is NP-hard, and computing neighbor-component order connectivity of a graph for an arbitrary threshold value is also NP-hard. We will look at the organizational structure of a unicycle graph which aids in modeling the vulnerability of a communication network. In order to assess the vulnerability of this network, we present a polynomial algorithm for computing the neighbor-component order connectivity of a unicycle. We also note that this algorithm covers the case of a threshold value of one and offers an algorithm for the domination number of the unicycle.