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Register a new userRelations between global forcing number and maximum anti-forcing number of a graph
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The global forcing number of a graph G is the minimal cardinality of an edge subset discriminating all perfect matchings of G, denoted by gf(G). For any perfect matching M of G, the minimal cardinality of an edge subset S in E(G)-M such that G-S has a unique perfect matching is called the anti-forcing number of M,denoted by af(G, M). The maximum anti-forcing number of G among all perfect matchings is denoted by Af(G). It is known that the maximum anti-forcing number of a hexagonal system equals the famous Fries number. We are interested in some comparisons between the global forcing number and the maximum anti-forcing number of a graph. For a bipartite graph G, we show that gf(G)is larger than or equal to Af(G). Next we mainly extend such result to non-bipartite graphs, which is the set of all graphs with a perfect matching which contain no two disjoint odd cycles such that their deletion results in a subgraph with a perfect matching. For any such graph G, we also have gf(G) is larger than or equal to Af(G) by revealing further property of non-bipartite graphs with a unique perfect matching. As a consequence, this relation also holds for the graphs whose perfect matching polytopes consist of non-negative 1-regular vectors. In particular, for a brick G, de Carvalho, Lucchesi and Murty [4] showed that G satisfying the above condition if and only if G is solid, and if and only if its perfect matching polytope consists of non-negative 1-regular vectors. Finally, we obtain tight upper and lower bounds on gf(G)-Af(G). For a connected bipartite graph G with 2n vertices, we have that 0 leq gf(G)-Af(G) leq 1/2 (n-1)(n-2); For non-bipartite case, -1/2 (n^2-n-2) leq gf(G)-Af(G) leq (n-1)(n-2).
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In this paper we compare the brushing number of a graph with the zero-forcing number of its line graph. We prove that the zero-forcing number of the line graph is an upper bound for the brushing number by constructing a brush configuration based on a zero-forcing set for the line graph. Using a similar construction, we also prove the conjecture that the zero-forcing number of a graph is no more than the zero-forcing number of its line graph; moreover we prove that the brushing number of a graph is no more than the brushing number of its line graph. All three bounds are shown to be tight.
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Let $G$ be a simple graph with $2n$ 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 $f(G)leq F(G)leq n-1$. Che and Chen (2011) proposed an open problem: how to characterize the graphs $G$ with $f(G)=n-1$. Later they showed that for bipartite graphs $G$, $f(G)=n-1$ if and only if $G$ is complete bipartite graph $K_{n,n}$. In this paper, we solve the problem for general graphs and obtain that $f(G)=n-1$ if and only if $G$ is a complete multipartite graph or $K^+_{n,n}$ ($K_{n,n}$ with arbitrary additional edges in the same partite set). For a larger class of graphs $G$ with $F(G)=n-1$ we show that $G$ is $n$-connected and a brick (3-connected and bicritical graph) except for $K^+_{n,n}$. In particular, we prove that the forcing spectrum of each such graph $G$ is continued by matching 2-switches and the minimum forcing numbers of all such graphs $G$ form an integer interval from $lfloorfrac{n}{2}rfloor$ to $n-1$.
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