A Note on Independent Sets in Trees
Copyright policy )A Note on Independent Sets in Trees
We give a simple graph-theoretical proof that the largest number of maximal independent vertex sets in a tree with n vertices is given by \[ m(T) = \begin{cases} 2^{k-1}+1 & \text{if \(n=2k,\)} \\ 2^ k & \text{if \(n=2k+1,\)} \end{cases} \] a result first proved by \textit{H. Wilf} [SIAM J. Algebraic Discrete Methods 7, 125-130 (1986; Zbl 0584.05024)]. We also characterize those trees achieving this maximum value. Finally we investigate some related problems.
- University of Pennsylvania United States
Extremal problems in graph theory, Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.), independent vertices, extremal graphs, Enumeration in graph theory, maximal independent vertex sets, tree
Extremal problems in graph theory, Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.), independent vertices, extremal graphs, Enumeration in graph theory, maximal independent vertex sets, tree
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