Defensive domination in proper interval graphs

The k-defensive domination problem, a variant of the classical problem on graphs, seeks minimum cardinality vertex set providing surjective defense against any attack vertices bounded by parameter k, where every under is defended distinct defender (which can be itself). has been shown to NP-complete for fixed k; if k part input, unlikely in co-NP. We present efficient algorithms solving this proper interval graphs with input. use greedy approach that takes advantage linear ordering endpoints intervals associated vertices. first algorithm based model and complexity O(n⋅k) graph n an at most second one improvement employs bubble representations realize improved O(n+B⋅logk) represented B bubbles.