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

Abstract

A set [Formula: see text] of a graph [Formula: see text] is called a connected power dominating set of [Formula: see text] if [Formula: see text], the subgraph induced by [Formula: see text], is connected and every vertex in the graph can be observed from [Formula: see text], following the two observation rules for power system monitoring: Rule [Formula: see text]: if [Formula: see text], then [Formula: see text] can observe itself and all its neighbors, and Rule [Formula: see text]: for an already observed vertex whose all neighbors except one are observed, then the only unobserved neighbor becomes observed as well. Given a graph [Formula: see text], Minimum Connected Power Domination is to find a connected power dominating set of minimum cardinality of [Formula: see text] and Decide Connected Power Domination is the decision version of Minimum Connected Power Domination. Decide Connected Power Domination is known to be NP -complete for general graphs. In this paper, we prove that Decide Connected Power Domination remains NP -complete for star-convex bipartite graphs, perfect elimination bipartite graphs and split graphs. This answers some open problems posed in [B. Brimkov, D. Mikesell and L. Smith, Connected power domination in graphs, J. Comb. Optim. 38(1) (2019) 292–315]. On the positive side, we show that Minimum Connected Power Domination 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 cannot be approximated within [Formula: see text] for any [Formula: see text] unless [Formula: see text], for bipartite graphs as well as for chordal graphs. Finally, we show that Minimum Connected Power Domination is APX -hard for bounded degree graphs.