3-Factor-criticality in domination critical graphs Original Research Article

A graph image is said to be k-image-critical if the size of any minimum dominating set of vertices is k, but if any edge is added to image the resulting graph can be dominated with image vertices. The structure of k-image-critical graphs remains far from completely understood when image. A graph image is factor-critical if image has a perfect matching for every vertex image and is bicritical if image has a perfect matching for every pair of distinct vertices image. More generally, a graph is said to be k-factor-critical if image has a perfect matching for every set S of k vertices in image. In three previous papers [N. Ananchuen, M.D. Plummer, Some results related to the toughness of 3-domination-critical graphs, Discrete Math. 272 (2003) 5–15; N. Ananchuen, M.D. Plummer, Matching properties in domination critical graphs, Discrete Math. 277 (2004) 1–13; N. Ananchuen, M.D. Plummer, Some results related to the toughness of 3-domination-critical graphs. II. Utilitas Math. 70 (2006) 11–32], we explored the toughness of 3-image-critical graphs and some of their matching properties. In particular, we obtained some properties which are sufficient for a 3-image-critical graph to be factor-critical and, respectively, bicritical. In the present work, we obtain similar results for k-factor-critical graphs when image.