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- W3169948011 abstract "Let $G$ be a simple graph with $2n$ vertices and a perfect matching. We denote by $f(G)$ and $F(G)$ the minimum and maximum forcing number of $G$, respectively. Hetyei obtained that the maximum number of edges of graphs $G$ with a unique perfect matching is $n^2$. We know that $G$ has a unique perfect matching if and only if $f(G)=0$. Along this line, we generalize the classical result to all graphs $G$ with $f(G)=k$ for $0leq kleq n-1$, and characterize corresponding extremal graphs as well. Hence we get a non-trivial lower bound of $f(G)$ in terms of the order and size. For bipartite graphs, we gain corresponding stronger results. Further, we obtain a new upper bound of $F(G)$. For bipartite graphs $G$, Che and Chen (2013) obtained that $f(G)=n-1$ if and only if $G$ is complete bipartite graph $K_{n,n}$. We completely characterize all bipartite graphs $G$ with $f(G)= n-2$." @default.
- W3169948011 created "2021-06-22" @default.
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- W3169948011 date "2021-06-16" @default.
- W3169948011 modified "2023-10-18" @default.
- W3169948011 title "Some tight bounds on the minimum and maximum forcing numbers of graphs" @default.
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- W3169948011 doi "https://doi.org/10.48550/arxiv.2106.09209" @default.
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