Improved Algorithms for k-Domination and Total k-Domination in Proper Interval Graphs

Abstract

Given a positive integer k, a k-dominating set in a graph G is a set of vertices such that every vertex not in the set has at least k neighbors in the set. A total k-dominating set, also known as a k-tuple total dominating set, is a set of vertices such that every vertex of the graph has at least k neighbors in the set. The problems of finding the minimum size of a k-dominating, resp. total k-dominating set, in a given graph, are referred to as k-domination, resp. total k-domination. These generalizations of the classical domination and total domination problems are known to be NP-hard in the class of chordal graphs, and, more specifically, even in the classes of split graphs (both problems) and undirected path graphs (in the case of total k-domination). On the other hand, it follows from recent work by Kang et al. (2017) that these two families of problems are solvable in time O(|V (G)| 6k+4 ) in the class of interval graphs. In this work, we develop faster algorithms for k-domination and total k-domination in the class of proper interval graphs. The algorithms run in time O(|V (G)| 3k ) for each fixed k ≥ 1 and are also applicable to the weighted case.

Total 2-domination of proper interval graphs

“…In this note we developed an O(m) time algorithm for the total 2-dominating set problem on proper interval graphs, improving the previous O(n 6 ) time algorithm by Chiarelli et al [2]. Both of these algorithms work by finding a shortest path on a weigthed digraph D. The main difference between them is that in our model the edges of D represent connected sets with a large diameter.…”

“…In turn, when G is an interval graph with n vertices, the problem is solvable in O(n 6k+4 ) time, as recently proven by Kang et al [4] (cf. [2]). Moreover, the time complexity can be reduced to O(n 3k ) when G belongs to the subclass of proper interval graphs [2].…”

“…[2]). Moreover, the time complexity can be reduced to O(n 3k ) when G belongs to the subclass of proper interval graphs [2].…”

“…Our algorithm, as well as the one by Chiarelli et al [2] and others, models the total 2-dominating problem as a shortest path problem on a weighted acyclic digraph D. The major difference is that, in the model by Chiarelli et al, each vertex of D represents a connected set of G with diameter at most 5. In turn, in our model each vertex represents different connected sets of varying diameters, that correspond to the weight of each outgoing edge and can be as high as Ω(n).…”

Tuple Domination on Graphs with the Consecutive-zeros Property

“…With a different approach, polynomial algorithms were recently provided for some variations of domination, say k-domination and total k-domination (for fixed k) for proper interval graphs [2].…”