Using the concept of minimum dominating set,maximum in-dependent set,new graph coloring theory based minimum connected dominating set algorithm was proposed to construct virtual backbone in wireless ad hoc networks.

In this paper, through the use of the disjunctive normal form and the principal disjunctive normal form, we propose some general algorithms concerning the grand minimal dominant set, the minimum dominant set, the minimal edge dominant set, and the minimum edge dominant set of graphs.

By establishing graph model of mine ventilation network,monitoring scope was applied as the theory base,flowing time between nodes was used as monitoring level,optimum methane sensor placement could be resolved with coverage as the selection standard based on Dijkstra arithmetic and minimal dominant set arithmetic.

The novel mult-level minimal dominating clustering arithmetic is the reprove of the random clustering arith-metic,we discussed the arithmetic of minimal dominating set,and developed a improved predigestion arithmetic of near minimal dominating set,reduced the workload of compute the minimal dominating set.

The minimal independent dominating set was settled by calculating maximal independent set,and with the set the algorithm and mathematical model of a virtual backbone network were provided.

Using the concept of minimum dominating set,maximum in-dependent set,new graph coloring theory based minimum connected dominating set algorithm was proposed to construct virtual backbone in wireless ad hoc networks.