Subexponential fixed-parameter algorithms for partial vector domination

Abstract

Given a graph G = (V, E) of order n and an n-dimensional non-negative vector d = (d(1), d(2),. .. , d(n)), called demand vector, the vector domination (resp., total vector domination) is the problem of finding a minimum S ⊆ V such that every vertex v in V \ S (resp., in V) has at least d(v) neighbors in S. The (total) vector domination is a generalization of many dominating set type problems, e.g., the (total) dominating set problem, the (total) k-dominating set problem (this k is different from the solution size), and so on, and subexponential fixed-parameter algorithms with respect to solution size for apex-minor-free graphs (so for planar graphs) are known. In this paper, we consider maximization versions of the problems; that is, for a given integer k, the goal is to find an S ⊆ V with size k that maximizes the total sum of satisfied demands. For these problems, we design subexponential fixed-parameter algorithms with respect to k for apex-minor-free

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

“…The total k-domination problem is NP-hard in the classes of split graphs [53], doubly chordal graphs [53], bipartite graphs [53], undirected path graphs [43], and bipartite planar graphs (for k ∈ {2, 3}) [1], and solvable in linear time in the class of graphs every block of which is a clique, a cycle, or a complete bipartite graph [43], and, more generally, in any class of graphs of bounded clique-width [19,50], and in polynomial time in the class of chordal bipartite graphs [53]. k-domination and total k-domination problems were also studied with respect to their (in)approximability properties, both in general [17] and in restricted graph classes [2], as well as from the parameterized complexity point of view [9,34].…”