This paper proposes a new polynomial time constant factor approximation algorithm for a more-a-decade-long open NP-hard problem, the minimum four-connected    $m$   -dominating set problem in unit disk graph (UDG) with any positive integer    $m \geq 1$    for the first time in the literature. We observe that it is difficult to modify the existing constant factor approximation algorithm for the minimum three-connected    $m$   -dominating set problem to solve the minimum four-connected    $m$   -dominating set problem in UDG due to the structural limitation of Tutte decomposition, which is the main graph theory tool used by Wang  et al.  to design their algorithm. To resolve this issue, we first reinvent a new constant factor approximation algorithm for the minimum three-connected    $m$   -dominating set problem in UDG and later use this algorithm to design a new constant factor approximation algorithm for the minimum four-connected    $m$   -dominating set problem in UDG.
['Wei Wang', 'Bei Liu', 'Dong-Hyun Kim', 'Deying Li', 'Jingyi Wang', 'Wei Gao']
A New Constant Factor Approximation to Construct Highly Fault-Tolerant Connected Dominating Set in Unit Disk Graph