Domination subdivision and domination multisubdivision numbers of graph

Abstract

The domination subdivision number sd(G) of a graph G is the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the domination number of G. It has been shown [9] that sd(T ) ≤ 3 for any tree T . We prove that the decision problem of the domination subdivision number is NP-complete even for bipartite graphs. For this reason we define the domination multisubdivision number of a nonempty graph G as a minimum positive integer k such that there exists an edge which must be subdivided k times to increase the domination number of G. We show that msd(G) ≤ 3 for any graph G. The domination subdivision number and the domination multisubdivision numer of a graph are incomparable in general case, but we show that for trees these two parameters are equal. We also determine domination multisubdivision number for some classes of graphs.

On domination multisubdivision number of unicyclic graphs

“…In this paper we continue the study of the domination multisubdivision number defined by Dettlaff, Raczek and Topp in [4]. Let msd(uv) be the minimum number of subdivisions of the edge uv such that γ(G) increase.…”

“…Domination multisubdivision number is well defined for all graphs with at least one edge. In [4] were also studied some complexity aspects regarding the domination subdivision and domination multisubdivision numbers of graphs. That is, there was studied the following decision problems.…”

“…and, Is msd(G) > 1? As a result, in [4], was obtained that these decision problems for the domination subdivision number, as well as for the domination multisubdivision number, are NP-complete even for bipartite graphs. In this sense, it is desirable to find or describe some families of graphs in which is possible to give the exact value for these parameters.…”

“…As it was proven in [4], msd(G) ∈ {1, 2, 3}. Interesting problem about graphs in which subdividing any single edge two times does not increase its dominating number arises.…”

“…What is their structure like? The class of all trees T with msd(T ) = 3 is already characterized and the next section sums up the results from [4] and [1] on this topic. Next we characterize all unicyclic graphs with the domination multisubdivision number equal to 3.…”