Complexity results for $k$-domination and $α$-domination problems and their variants

Abstract

Let G = (V, E) be a simple and undirected graph. For some integer k 1, a set D ⊆ V is said to be a k-dominating set in G if every vertex v of G outside D has at least k neighbors in D. Furthermore, for some real number α with 0 < α 1, a set D ⊆ V is called an α-dominating set in G if every vertex v of G outside D has at least α×dv neighbors in D, where dv is the degree of v in G. The cardinality of a minimum k-dominating set and a minimum α-dominating set in G is said to be the k-domination number and the α-domination number of G, respectively. In this paper, we present some approximability and inapproximability results on the problem of finding of k-domination number and α-domination number of some classes of graphs. Moreover, we introduce a generalization of α-dominating set which we call f -dominating set. Given a function f : N → R, where N = {1, 2, 3, . . .}, a set D ⊆ V is said to be an f -dominating set in G if every vertex v of G outside D has at least f (dv) neighbors in D. We prove NP-hardness of the problem of finding of a minimum f -dominating set in G, for a large family of functions f .

New algorithms for weighted k-domination and total k-domination problems in proper interval graphs

“…All these intervals belong to vertices from component C 1 , and therefore to I s . This establishes property (2). A similar argument shows that if s ∈ B, then in order to make sure that the vertex corresponding to min(s ) has at least k neighbors in U , interval min(s ) must intersect the k intervals associated with s immediately following min(s ) in the sequence.…”

“…Several results on the complexity of k-domination and total k-domination problems were established in the literature. For every k, the k-domination problem is NP-hard in the classes of bipartite graphs [2] and split graphs [45]. The problem is solvable in linear time in the class of graphs every block of which is a clique, a cycle or a complete bipartite graph (including trees, block graphs, cacti, and block-cactus graphs) [45], and, more generally, in any class of graphs of bounded clique-width [21,53] (see also [17]).…”

“…The problem is solvable in linear time in the class of graphs every block of which is a clique, a cycle, or a complete bipartite graph [46], and, more generally, in any class of graphs of bounded clique-width [21,53], and in polynomial time in the class of chordal bipartite graphs [56]. k-domination and total k-domination problems were also studied with respect to their (in)approximability properties, both in general [18] and in restricted graph classes [2], as well as from the parameterized complexity point of view [9,37].…”

“…Note that since neither of I 0 and I n+1 is a big node, (I 0 , I n+1 ) ∈ E 1 ; moreover, (I 0 , I n+1 ) ∈ E 0 since condition (2) in the definition of an arc in E 0 fails. Therefore, (I 0 , I n+1 ) ∈ E(D t k ) and P has at least 3 nodes.…”

“…Suppose first that W i ∩ S = ∅. Since every arc in E 1 connects a pair of big nodes, we infer that W i = {s} for some s ∈ S. Using property (2) in the definition of a small node, we infer that G[U i ] is connected. Therefore, since no edge of G connects a vertex in U i with a vertex in U j for j = i, we infer that G[U i ] is a component of G[U ], as claimed.…”