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independent sets; circuit
Let the number of $k$-element sets of independent vertices and edges of a graph $G$ be denoted by $n(G,k)$ and $m(G,k)$, respectively. It is shown that the graphs whose every component is a circuit are the only graphs for which the equality $n(G,k)=m(G,k)$ is satisfied for all values of $k$.
[1] I. Gutman: On independent vertices and edges of a graph. Topics in Combinatorics and Graph Theory (R. Bodendiek and R. Henn, eds.). Physica-Verlag, Heidelberg, 1990, pp. 291-296. MR 1100048 | Zbl 0697.05038
[2] F. Harary: Graph Theory. Addison-Wesley, Reading, MA, 1969. MR 0256911 | Zbl 0196.27202
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