Upper Bounds on the Paired Domination Subdivision Number of a Graph

A Note on the Paired-Domination Subdivision Number of Trees

“…It should also be noted that Favaron et al [18] conjectured that sd γ pr (G) ≤ n − 1 for all connected graphs of order n. In connection with this conjecture, Egawa et al [20] proved that for every connected graph G of order n ≥ 4, sd γ pr (G) ≤ 2n − 5. Moreover, if further G has an edge uv such that u and v are not partners in any γ pr (G)-set, then sd γ pr (G) ≤ n − 1.…”

“…It is worth noting that it has recently been shown by Amjadi and Chellali [19] that the problem of computing the paired-domination subdivision number is NP-hard for bipartite graphs. The paired-domination subdivision number has been further studied by several authors (see [20][21][22]).…”

On the Paired-Domination Subdivision Number of Trees

“…If G is a connected graph of order at least 3, Favaron et al [16] asked whether it is true that for any edge e / ∈ E(G), sd γ pr (G + e) ≤ sd γ pr (G). Egawa et al [18] gave a negative answer to this question. However, they proved the question in the affirmative if the following additional condition is added: γ pr (G + e) < γ pr (G) for every e / ∈ E(G).…”

“…In this paper, we study the paired-domination subdivision number of trees, which was introduced by Favaron et al in [16] and has been further studied in [17][18][19][20][21]. The paireddomination subdivision number sd γ pr (G) of a graph G is the minimum number of edges that must be subdivided (where each edge in G can be subdivided at most once) in order to increase the paired-domination number of G.…”