Effect of edge-subdivision on vertex-domination in a graph

Abstract

Let G be a graph with ∆(G) > 1. It can be shown that the domination number of the graph obtained from G by subdividing every edge exactly once is more than that of G. So, let ξ(G) be the least number of edges such that subdividing each of these edges exactly once results in a graph whose domination number is more than that of G. The parameter ξ(G) is called the subdivision number of G. This notion has been introduced by S. Arumugam and S. Velammal. They have conjectured that for any graph G with ∆(G) > 1, ξ(G) ≤ 3. We show that the conjecture is false and construct for any positive integer n ≥ 3, a graph G of order n with ξ(G) > 1 3 log 2 n. The main results of this paper are the following: (i) For any connected graph G with at least three vertices, ξ(G) ≤ γ(G) + 1 where γ(G) is the domination number of G. (ii) If G is a connected graph of sufficiently large order n, then ξ(G) ≤ 4√ n ln n + 5.

Coronas and Domination Subdivision Number of a Graph

“…Since the domination number of the graph K 2 does not increase when its edge is subdivided, we consider the subdivision numbers for connected graphs of order at least 3. The domination subdivision number was defined by Velammal [8] and since then it has been widely studied, see [2,3,4,5,6] to mention just a few.…”

Edge subdivision and edge multisubdivision versus some domination related parameters in generalized corona graphs

“…Since the paired multisubdivision number for any graph is equal either 1, 2, 3 or 4, it suffices to justify that subdividing any edge of G H, where H = {K 1 } with three vertices does not increase its paired domination number. Subdivide any uu edge with three new vertices, where u ∈ V (G) and u is the neighbor of u with degree one in G H. Then the set formed by the vertices of the graph induced by a maximum matching in G not containing u, altogether with two adjacent subdivision vertices, where one of them is a support vertex in (G H) uu , 3 and with unmatched vertices of V (G) − {u} paired with any neighboring vertex of V (G H) − V (G), is a paired dominating set of (G H) uu ,3 of cardinality γ pr (G H). Now subdivide any edge uw ∈ E(G H) with three subdivision vertices, where u, w ∈ V (G).…”

“…This parameter was studied in trees by Aram, Sheikholeslami and Favaron [1] and also by Benecke and Mynhardt [2]. General bounds and properties has been studied for example by Haynes, Hedetniemi 576 Magda Dettlaff, Joanna Raczek, and Ismael G. Yero and Hedetniemi [10], by Bhattacharya and Vijayakumar [3], by Favaron, Haynes and Hedetniemi [5] and by Favaron, Karami and Sheikholeslami [6].…”

Total domination and total domination subdivision number of a graph and its complement

“…Similar definitions exist for the domination number (G) and the domination subdivision number sd (G) and, when G is connected, for the connected domination number c (G) and the connected domination subdivision number sd c (G). The first of these was introduced for the domination number in Velammal's thesis [14] and since this time many results have been obtained on the three parameters sd , sd t , sd c (see for instance [1,[4][5][6][8][9][10][11]). Since the total domination number of the graph K 2 does not change when its only edge is subdivided, in the study of the total domination subdivision number of a connected graph G, we assume that G has order n 3.…”