A graph is total domination edge-critical if the addition of any edge decreases the total domination number, while a graph with minimum degree at least two is total domination vertex-critical if the removal of any vertex decreases the total domination number. A 3 t EC graph is a total domination edge-critical graph with total domination number 3 and a 3 t VC graph is a total domination vertex-critical graph with total domination number 3. A graph G is factor-critical if G − v has a perfect matching for every vertex v in G. In this paper, we show that every 3 t EC graph of even order has a perfect matching, while every 3 t EC graph of odd order with no cut-vertex is factor-critical. We also show that every 3 t VC graph of even order that is K 1,7-free has a perfect matching, while every 3 t VC graph of odd order that is K 1,6-free is factor-critical. We show that these results are tight in the sense that there exist 3 t VC graphs of even order with no perfect matching that are K 1,8-free and 3 t VC graphs of odd order that are K 1,7-free but not factor-critical.
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机译：如果添加任何边会减少总控制数，则图是关键的总控制边；如果移除任意顶点会减少总控制数，则具有最小度数至少为2的图是总控制关键点。 3 t EC图是总支配数为3的总支配边临界图，3 t VC图是总支配数为总的支配顶点临界图3.如果G − v对G中的每个顶点v都具有完美匹配，则图G是关键因子。在本文中，我们证明了每3个偶数阶的 t EC图都具有完美匹配，而每3个 t 奇数阶EC图（无割顶点）都是关键因子。我们还显示，无偶数K 1,7 的每3 t VC图具有完美匹配，而每3 t 无K 1,6 的奇数阶VC图是关键因子。我们表明，在存在3个 t 偶数阶VC图且没有完美匹配的情况下，这些结果是紧密的，它们没有K 1,8 和3 t 奇数次的VC图，无K 1,7 ，但不是关键因子。
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