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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.)
##### Keywords:
decycling number; independence number; cycle rank; margin number
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