Algorithmic Complexity and Bounds for Domination Subdivision Numbers of Graphs

Abstract

Let G=V,E be a simple graph. A subset D⊆V is a dominating set if every vertex not in D is adjacent to a vertex in D. The domination number of G, denoted by γG, is the smallest cardinality of a dominating set of G. The domination subdivision number sdγG of G is the minimum number of edges that must be subdivided (each edge can be subdivided at most once) in order to increase the domination number. In 2000, Haynes et al. showed that sdγG≤dGv+dGv−1 for any edge uv∈EG with dGu≥2 and dGv≥2 where G is a connected graph with order no less than 3. In this paper, we improve the above bound to sdγG≤dGu+dGv−NGu∩NGv−1, and furthermore, we show the decision problem for determining whether sdγG=1 is NP-hard. Moreover, we show some bounds or exact values for domination subdivision numbers of some graphs.