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Graphs whose vertex set can be partitioned into a total dominating set and an independent dominating set

Haynes Teresa W., Henning Michael A.

Opuscula Mathematica

2024

Vol. 44, no. 4

543--563

A graph G whose vertex set can be partitioned into a total dominating set and an independent dominating set is called a TI-graph. We give constructions that yield infinite families of graphs that are TI-graphs, as well as constructions that yield infinite families of graphs that are not TI-graphs. We study regular graphs that are TI-graphs. Among other results, we prove that all toroidal graphs are TI-graphs.

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

Dettlaff M., Raczek J., Yero I. G.

Opuscula Mathematica

2016

Vol. 36, no. 5

575--588

Given a graph G = (V, E), the subdivision of an edge e = uv ∈ E(G) means the substitution of the edge e by a vertex x and the new edges ux and xv. The domination subdivision number of a graph G is the minimum number of edges of G which must be subdivided (where each edge can be subdivided at most once) in order to increase the domination number. Also, the domination multisubdivision number of G is the minimum number of subdivisions which must be done in one edge such that the domination number increases. Moreover, the concepts of paired domination and independent domination subdivision (respectively multisubdivision) numbers are denned similarly. In this paper we study the domination, paired domination and independent domination (subdivision and multisubdivision) numbers of the generalized corona graphs.