3-factor-criticality in domination critical graphs (Q2462368)

For an integer \(k\geq 2\) a graph \(G\) is \(k\)-\(\gamma\)-critical if the domination number \(\gamma(G)\) of \(G\) is \(k\) and \(\gamma(G+e) = k-1\) for every edge \(e \not\in E(G)\). For an integer \(t \geq 1\) a graph \(G\) is \(t\)-factor-critical if \(G-S\) has a perfect matching for every set \(S\) of \(t\) vertices of \(G\). The authors show that if \(G\) is a \(4\)-connected \(3\)-\(\gamma\)-critical graph of odd order and \(\delta(G) \geq 5\), then \(G\) is \(3\)-factor-critical. They also show that if \(G\) is a claw-free graph of odd order that is \(3\)-\(\gamma\)-critical, \(3\)-connected and has \(\delta(G) \geq 4\), then \(G\) is \(3\)-factor-critical.