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On minimum intersections of certain secondary dominating sets in graphs

Kosiorowska Anna, Michalski Adrian, Włoch Iwona

Opuscula Mathematica

2023

Vol. 43, no. 6

813--827

In this paper we consider secondary dominating sets, also named as (1,k)-dominating sets, introduced by Hedetniemi et al. in 2008. In particular, we study intersections of the (1, 1)-dominating sets and proper (1, 2)-dominating sets. We introduce (1,2)-intersection index as the minimum possible cardinality of such intersection and determine its value for some classes of graphs.

Trees whose 2-domination subdivision number is 2

Atapour M., Sheikholeslami S. M., Khodkar A.

Opuscula Mathematica

2012

Vol. 32, no. 3

423-437

A set S of vertices in a graph G = (V,E) is a 2-dominating set if every vertex of V \ S is adjacent to at least two vertices of S. The 2-domination number of a graph G, denoted by γ2(G), is the minimum size of a 2-dominating set of G. The 2-domination subdivision number sdγ2 (G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the 2-domination number. The authors have recently proved that for any tree T of order at least 3, 1 ≤ sdγ2 (T ) ≤ 2. In this paper we provide a constructive characterization of the trees whose 2-domination subdivision number is 2.