In this paper, we will study a special Connected Dominating Set (CDS) problem - between any two nodes in a network, there exists at least one shortest path, all of whose intermediate nodes should be included in a special CDS, named Minimum rOuting Cost CDS (MOC-CDS). Therefore, routing by MOC-CDS can guarantee that each routing path between any pair of nodes is also the shortest path in the network. Thus, energy consumption and delivery delay can be reduced greatly. CDS has been studied extensively in Unit Disk Graph (UDG) or Disk Graph (DG). However, nodes in networks may have different transmission ranges and some communications may be obstructed by obstacles. Therefore, we model network as a bidirectional general graph in this paper. We prove that constructing a minimum MOC-CDS in general graph is NP-hard. We also prove that there does not exist a polynomial-time approximation algorithm for constructing a minimum MOC-CDS with performance ratio ρlnδ, where ρ is an arbitrary positive number (ρ < 1) and δ is the maximum node degree in network. We propose a distributed heuristic algorithm (called as FlagContest) for constructing MOC-CDS with performance ratio (1 - ln2) + 2lnδ. Through extensive simulations, we show that the results of FlagContest is within the upper bound proved in this paper. Simulations also demonstrate that the average length of routing paths through MOC-CDS reduces greatly compared to regular CDSs.