Defensive Domination in Proper Interval Graphs
T{\i}naz Ekim, Arthur Farley, Andrzej Proskurowski, Mordechai, Shalom

TL;DR
This paper introduces efficient algorithms for solving the NP-complete $k$-defensive domination problem on proper interval graphs, leveraging interval orderings and bubble representations for improved computational complexity.
Contribution
It presents two novel algorithms that efficiently solve the $k$-defensive domination problem on proper interval graphs, with complexity improvements over previous methods.
Findings
First algorithm runs in ${ m O}(n imes k)$ time.
Second algorithm improves complexity to ${ m O}(n + | ext{B}| imes ext{log} k)$.
Algorithms exploit linear orderings and bubble representations of proper interval graphs.
Abstract
-defensive domination, a variant of the classical domination problem on graphs, seeks a minimum cardinality vertex set providing a surjective defense against any attack on vertices of cardinality bounded by a parameter . The problem has been shown to be NP-complete} for fixed ; if is part of the input, the problem is not even in NP. We present efficient algorithms solving this problem on proper interval graphs with part of the input. The algorithms take advantage of the linear orderings of the end points of the intervals associated with vertices to realize a greedy approach to solution. The first algorithm is based on the interval model and has complexity for a graph on vertices. The second one is an improvement of the first and employs bubble representations of proper interval graph to realize an improved complexity of ${\cal O}(n+ \vert{\cal…
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Taxonomy
TopicsComplexity and Algorithms in Graphs · Advanced Graph Theory Research · Logic, Reasoning, and Knowledge
