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New formulae for the decycling number of graphs. (English) Zbl 1401.05074
Summary: A set \(S\) of vertices of a graph \(G\) is called a decycling set if \(G-S\) is acyclic. The minimum order of a decycling set is called the decycling number of \(G\), and denoted by \(\nabla(G)\). Our results include: (a) For any graph \(G\), \[ \nabla(G)=n-\max_{T}\{\alpha(G-E(T))\}, \] where \(T\) is taken over all the spanning trees of \(G\) and \(\alpha(G - E(T))\) is the independence number of the co-tree \(G - E(T)\). This formula implies that computing the decycling number of a graph \(G\) is equivalent to finding a spanning tree in \(G\) such that its co-tree has the largest independence number. Applying the formula, the lower bounds for the decycling number of some (dense) graphs may be obtained. (b) For any decycling set \(S\) of a \(k\)-regular graph \(G\), \[ |S|=\frac{1}{k-1}(\beta(G)+m(S)), \] where \(\beta(G) = |E(G)|-|V (G)|+1\) and \(m(S) = c+|E(S)|-1\), \(c\) and \(|E(S)|\) are, respectively, the number of components of \(G - S\) and the number of edges in \(G[S]\). Hence \(S\) is a \(\nabla\)-set if and only if \(m(S)\) is minimum, where \(\nabla\)-set denotes a decycling set containing exactly \(\nabla(G)\) vertices of \(G\). This provides a new way to locate \(\nabla(G)\) for \(k\)-regular graphs \(G\). (c) 4-regular graphs \(G\) with the decycling number \(\nabla(G)\left\lceil \frac{\beta(G)}{3}\right\rceil\) are determined.
MSC:
05C07 Vertex degrees
05C38 Paths and cycles
05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
05C42 Density (toughness, etc.)
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