On 3-?t ? t -vertex critical graphs of diameter three
A total dominating set of a graph G=(V,E) G = ( V , E ) with no isolated vertex is a set S?V S ? V such that every vertex is adjacent to a vertex in S S . The minimum cardinality of a total dominating set of G G is the total domination number ?t(G) ? t ( G ) of G G . Let k?3 k ? 3 be an integer. A graph G G with no isolated vertex is k k -?t ? t -vertex critical if ?t(G)=k ? t ( G ) = k and ?t(G?v)<k ? t ( G ? v ) < k for every vertex v v of G G that is not adjacent to a vertex of degree one. In this note we study 3-?t ? t -vertex critical graphs of diameter three. We prove that such graphs have order at least 10, and we characterize all 3-?t ? t -vertex critical graphs of order 10. Moreover, we show that for every even number n?10 n ? 10 , there is a 3-?t ? t -vertex critical graph of order n n .
diameter total domination vertex critical