Hardness Results of Connected Power Domination for Bipartite Graphs and Chordal Graphs

A set D subset of V of a graph G = (V, E) is called a connected power dominating set of G if G[D], the subgraph induced by D, is connected and every vertex in the graph can be observed from D, following the two observation rules for power system monitoring: Rule 1: if v is an element of D, then v can observe itself and all its neighbors, and Rule 2: for an already observed vertex whose all neighbors except one are observed, then the only unobserved neighbor becomes observed as well. MINIMUM CONNECTED POWER DOMINATION PROBLEM is to find a connected power dominating set of minimum cardinality of a given graph G and Decide CONNECTED POWER DOMINATION PROBLEM is the decision version of MINIMUM CONNECTED POWER DOMINATION PROBLEM. DECIDE CONNECTED POWER DOMINATION PROBLEM is known to be NP-complete for general graphs. In this paper, we strengthen this result by proving that DECIDE CONNECTED POWER DOMINATION PROBLEM remains NP-complete for perfect elimination bipartite graph, a proper subclass of bipartite graphs, and split graphs, a proper subclass of chordal graphs. On the positive side, we show that MINIMUM CONNECTED POWER DOMINATION PROBLEM is polynomial-time solvable for chain graphs, a proper subclass of perfect elimination bipartite graph, and for threshold graphs, a proper subclass of split graphs. Further, we show that MINIMUM CONNECTED POWER DOMINATION PROBLEM cannot be approximated within (1 - epsilon) ln vertical bar V vertical bar for any epsilon > 0 unless P = NP, for bipartite graphs as well as for chordal graphs.