Maximizing the Minimum and Maximum Forcing Numbers of Perfect Matchings of Graphs

• Published in: Acta mathematica Sinica. English series Vol. 39; no. 7; pp. 1289 - 1304
• Main Authors: Liu, Qian Qian, Zhang, He Ping
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.07.2023; Springer Nature B.V
• Subjects: Apexes; Graph theory; Graphs; Integers; Matching; Mathematics; Mathematics and Statistics; Switches; 05C75; complete multipartite graph; maximum forcing number; forcing spectrum; 05C70; Perfect matching; 05C35; minimum forcing number
• DOI: 10.1007/s10114-023-1020-6
• ISSN: 1439-8516

Let G be a simple graph with 2 n vertices and a perfect matching. The forcing number f ( G, M ) of a perfect matching M of G is the smallest cardinality of a subset of M that is contained in no other perfect matching of G . Among all perfect matchings M of G , the minimum and maximum values of f ( G, M ) are called the minimum and maximum forcing numbers of G , denoted by f ( G ) and F ( G ), respectively. Then f ( G ) ≤ F ( G ) ≤ n − 1. Che and Chen (2011) proposed an open problem: how to characterize the graphs G with f ( G ) = n − 1. Later they showed that for a bipartite graph G, f ( G )= n − 1 if and only if G is complete bipartite graph K n,n . In this paper, we completely solve the problem of Che and Chen, and show that f ( G )= n − 1 if and only if G is a complete multipartite graph or a graph obtained from complete bipartite graph K n,n by adding arbitrary edges in one partite set. For all graphs G with F ( G ) = n − 1, we prove that the forcing spectrum of each such graph G forms an integer interval by matching 2-switches and the minimum forcing numbers of all such graphs G form an integer interval from ⌊ n 2 ⌋ to n − 1.