Algorithmic Aspects of Disjunctive Domination in Graphs

Abstract

For a graph G = (V, E), a set D ⊆ V is called a disjunctive dominating set of G if for every vertex v ∈ V \ D, v is either adjacent to a vertex of D or has at least two vertices in D at distance 2 from it. The cardinality of a minimum disjunctive dominating set of G is called the disjunctive domination number of graph G, and is denoted by γ. Given a positive integer k and a graph G, the DISJUNCTIVE DOMINATION DECISION PROBLEM (DDDP) is to decide whether G has a disjunctive dominating set of cardinality at most k. In this article, we first propose a linear time algorithm for MDDP in proper interval graphs. Next we tighten the NP-completeness of DDDP by showing that it remains NP-complete even in chordal graphs. We also propose a (ln(∆ 2 + ∆ + 2) + 1)-approximation algorithm for MDDP in general graphs and prove that MDDP can not be approximated within (1 − ) ln(|V |) for any > 0 unless NP ⊆ DTIME(|V | O(log log |V |) ). Finally, we show that MDDP is APX-complete for bipartite graphs with maximum degree 3.

Relating domination, exponential domination, and porous exponential domination

“…There is not a single complexity result, and it is even unknown whether the porous exponential domination number of a given subcubic tree can be determined efficiently. Partly motivated by such difficulties, Goddard et al define [7] disjunctive domination (see also [10,11,13]), which keeps the exponential decay of the influence but only considers distances one and two. Further related parameters known as influence and total influence [6] also have unknown complexity even for trees [9].…”