Lucas C. van der Merwe

The diameter of total domination vertex critical graphs

Abstract A graph G with no isolated vertex is total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G - v is less than the total domination number of G . These graphs we call γ t -critical. If such a graph G has total domination number k , we call it k - γ t -critical. We characterize the connected graphs with minimum degree one that are γ t -critical and we obtain sharp bounds on their maximum diameter. We calculate the maximum diameter of a k - γ t -critical graph for k ⩽ 8 and provide an example which shows that the maximum diameter is in general at least 5 k / 3 - O ( 1 ) .

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.

Domination Subdivision Numbers

A set S of vertices of a graph G = (V,E) is a dominating set if every vertex of V − S is adjacent to some vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set of G, and the domination subdivision number sdγ(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 domination number. Arumugam conjectured that 1 ≤ sdγ(G) ≤ 3 for any graph G. We give a counterexample to this conjecture. On the other hand, we show that sdγ(G) ≤ γ(G)+1 for any graph G without isolated vertices, and give constant upper bounds on sdγ(G) for several families of graphs.

Progress on the Murty---Simon Conjecture on diameter-2 critical graphs: a survey

A graph

A maximum degree theorem for diameter-2-critical graphs

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Restrained domination in unicyclic graphs

G is diameter

Connected Domination Stable Graphs Upon Edge Addition

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