A Sufficient and Necessary Condition for the Forcing Number of a Bipartite Graph Being Equal to the Minimum Number of Trailing Vertices

摘要

Riddle [15] showed that the forcing number of a bipartite graph is bounded blow by the minimum number of trailing vertices of the ordering of a color set. In the present work, we improve the trailing-vertex method by Riddle and obtain a necessary condition for the matching forcing number of a bipartite graph being equal to a given natural number k; furthermore, we give a sufficient and necessary condition for the minimum forcing number of bipartite graph being equal to the minimum number of trailing vertices of all standard orderings of a color set.