Total edge critical graphs with leaves

Total Domination Edge Critical Graphs with Maximum Diameter

Denote the total domination number of a graph G by γt(G). A graph G is said to be total domination edge critical, or simply γtcritical, if γt(G + e) < γt(G) for each edge e ∈ E(G). For 3t-critical graphs G, that is, γt-critical graphs with γt(G) = 3, the diameter of G is either 2 or 3. We characterise the 3t-critical graphs G with diam G = 3.

A characterization of locating-total domination edge critical graphs

For a graph G = (V,E) without isolated vertices, a subset D of vertices of V is a total dominating set (TDS) of G if every vertex in V is adjacent to a vertex in D. The total domination number γt(G) is the minimum cardinality of a TDS of G. A subset D of V which is a total dominating set, is a locating-total dominating set, or just a LTDS of G, if for any two distinct vertices u and v of V (G) ...

Characterization of total restrained domination edge critical unicyclic graphs

A graph with no isolated vertices is edge critical with respect to total restrained domination if for any non-edge e of G, the total restrained domination number of G+ e is less than the total restrained domination number of G. We call these graphs γtr-edge critical. In this paper, we characterize all γtr-edge critical unicyclic graphs.

On the total restrained domination edge critical graphs

Let G = (V, E) be a graph. A set D ⊆ V is a total restrained dominating set of G if every vertex in V has a neighbor in D and every vertex in V −D has a neighbor in V −D. The cardinality of a minimum total restrained dominating set in G is the total restrained domination number of G. In this paper, we define the concept of total restrained domination edge critical graphs, find a lower bound for...

On the edge and total GA indices of some graphs