Perfect Matchings in Total Domination Critical Graphs

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_tEC graph is a total domination edge-critical graph with total domination number 3 and a 3_tVC 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_tEC graph of even order has a perfect matching, while every 3_tEC graph of odd order with no cut-vertex is factor-critical. We also show that every 3_tVC graph of even order that is K_1,7-free has a perfect matching, while every 3_tVC 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_tVC graphs of even order with no perfect matching that are K_1,8-free and 3_tVC graphs of odd order that are K_1,7-free but not factor-critical.