Solving Minimum Connected Dominating Set on Proper Interval Graph

To solve connected dominating problem, it is necessary to find minimum connected dominating set (MCDS for short). However, to find MCDS is NP-hardness. So, a model of graphs called interval graph was constructed from nodes of related network. Two greedy algorithms with linear (or polynomial time) were used to find MCDS on proper interval graph (or interval graph), and have 1 approximation ratio on the graphs. And spanning trees were constructed and used to validate the correctness and effectiveness of corresponding algorithms.