Graph Theory With Applications
Graph Theory With Applications
Note: On total domination vertex critical graphs of high connectivity
Discrete Applied Mathematics
On the existence problem of the total domination vertex critical graphs
Discrete Applied Mathematics
Note: On 3-γt-vertex critical graphs of diameter three
Discrete Applied Mathematics
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A vertex subset S of graph G is a total dominating set of G if every vertex of G is adjacent to a vertex in S. For a graph G with no isolated vertex, the total domination number of G, denoted by @c"t(G), is the minimum cardinality of a total dominating set. A total dominating set of cardinality @c"t(G) is called a @c"t(G)-set. 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. We call these graphs @c"t-critical. If such a graph G has total domination number k, then we call it k-@c"t-critical. In this note we study 4-@c"t-critical connected graphs G of diameter two. We prove that such graphs with minimum at least two have order at least 10, and we characterize all 4-@c"t-critical connected graphs of order 10 with maximum degree 5. Moreover, we obtain some 4-@c"t-critical connected graphs of order 10 with maximum degree 4 and for any integer k=2, n=3k+5, there exists a 4-@c"t-critical graph G of order n with diam(G)=2.