The maximum number of maximum generalized 4‐independent sets in trees
A generalized k $k$‐independent set is a set of vertices such that the induced subgraph contains no trees with k $k$‐vertices, and the generalized k $k$‐independence number is the cardinality of a maximum k $k$‐independent set in G $G$. Zito proved that the maximum number of maximum generalized 2‐in...
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Published in | Journal of graph theory Vol. 107; no. 2; pp. 359 - 380 |
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Abstract | A generalized k $k$‐independent set is a set of vertices such that the induced subgraph contains no trees with k $k$‐vertices, and the generalized k $k$‐independence number is the cardinality of a maximum k $k$‐independent set in G $G$. Zito proved that the maximum number of maximum generalized 2‐independent sets in a tree of order n $n$ is 2n−32 ${2}^{\frac{n-3}{2}}$ if n $n$ is odd, and 2n−22+1 ${2}^{\frac{n-2}{2}}+1$ if n $n$ is even. Tu et al. showed that the maximum number of maximum generalized 3‐independent sets in a tree of order n $n$ is 3n3−1+n3+1 ${3}^{\frac{n}{3}-1}+\frac{n}{3}+1$ if n≡0 (mod 3) $n\equiv 0\unicode{x02007}(\text{mod 3})$, and 3n−13−1+1 ${3}^{\frac{n-1}{3}-1}+1$ if n≡1 (mod 3) $n\equiv 1\unicode{x02007}(\text{mod 3})$, and 3n−23−1 ${3}^{\frac{n-2}{3}-1}$ if n≡2 (mod 3) $n\equiv 2\unicode{x02007}(\text{mod 3})$ and they characterized all the extremal graphs. Inspired by these two nice results, we establish four structure theorems about maximum generalized k $k$‐independent sets in a tree for a general integer k $k$. As applications, we show that the maximum number of generalized 4‐independent sets in a tree of order n (n≥4) $n\unicode{x02007}(n\ge 4)$ is
4n−44+(n4+1)(n8+1),n≡0 (mod 4),4n−54+n−14+1,n≡1 (mod 4),4n−64+n−24,n≡2 (mod 4),4n−74,n≡3 (mod 4) $\left\{\begin{array}{cc}{4}^{\frac{n-4}{4}}+(\frac{n}{4}+1)(\frac{n}{8}+1), & n\equiv 0\unicode{x02007}(\,\text{mod 4}\,),\\ {4}^{\frac{n-5}{4}}+\frac{n-1}{4}+1, & n\equiv 1\unicode{x02007}(\,\text{mod 4}\,),\\ {4}^{\frac{n-6}{4}}+\frac{n-2}{4}, & n\equiv 2\unicode{x02007}(\,\text{mod 4}\,),\\ {4}^{\frac{n-7}{4}}, & n\equiv 3\unicode{x02007}(\,\text{mod 4}\,)\end{array}\right.$ and we also characterize the structure of all extremal trees with the maximum number of maximum generalized 4‐independent sets. |
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AbstractList | Abstract A generalized ‐independent set is a set of vertices such that the induced subgraph contains no trees with ‐vertices, and the generalized ‐independence number is the cardinality of a maximum ‐independent set in . Zito proved that the maximum number of maximum generalized 2‐independent sets in a tree of order is if is odd, and if is even. Tu et al. showed that the maximum number of maximum generalized 3‐independent sets in a tree of order is if , and if , and if and they characterized all the extremal graphs. Inspired by these two nice results, we establish four structure theorems about maximum generalized ‐independent sets in a tree for a general integer . As applications, we show that the maximum number of generalized 4‐independent sets in a tree of order is and we also characterize the structure of all extremal trees with the maximum number of maximum generalized 4‐independent sets. A generalized k $k$‐independent set is a set of vertices such that the induced subgraph contains no trees with k $k$‐vertices, and the generalized k $k$‐independence number is the cardinality of a maximum k $k$‐independent set in G $G$. Zito proved that the maximum number of maximum generalized 2‐independent sets in a tree of order n $n$ is 2n−32 ${2}^{\frac{n-3}{2}}$ if n $n$ is odd, and 2n−22+1 ${2}^{\frac{n-2}{2}}+1$ if n $n$ is even. Tu et al. showed that the maximum number of maximum generalized 3‐independent sets in a tree of order n $n$ is 3n3−1+n3+1 ${3}^{\frac{n}{3}-1}+\frac{n}{3}+1$ if n≡0 (mod 3) $n\equiv 0\unicode{x02007}(\text{mod 3})$, and 3n−13−1+1 ${3}^{\frac{n-1}{3}-1}+1$ if n≡1 (mod 3) $n\equiv 1\unicode{x02007}(\text{mod 3})$, and 3n−23−1 ${3}^{\frac{n-2}{3}-1}$ if n≡2 (mod 3) $n\equiv 2\unicode{x02007}(\text{mod 3})$ and they characterized all the extremal graphs. Inspired by these two nice results, we establish four structure theorems about maximum generalized k $k$‐independent sets in a tree for a general integer k $k$. As applications, we show that the maximum number of generalized 4‐independent sets in a tree of order n (n≥4) $n\unicode{x02007}(n\ge 4)$ is4n−44+(n4+1)(n8+1),n≡0 (mod 4),4n−54+n−14+1,n≡1 (mod 4),4n−64+n−24,n≡2 (mod 4),4n−74,n≡3 (mod 4) $\left\{\begin{array}{cc}{4}^{\frac{n-4}{4}}+(\frac{n}{4}+1)(\frac{n}{8}+1), & n\equiv 0\unicode{x02007}(\,\text{mod 4}\,),\\ {4}^{\frac{n-5}{4}}+\frac{n-1}{4}+1, & n\equiv 1\unicode{x02007}(\,\text{mod 4}\,),\\ {4}^{\frac{n-6}{4}}+\frac{n-2}{4}, & n\equiv 2\unicode{x02007}(\,\text{mod 4}\,),\\ {4}^{\frac{n-7}{4}}, & n\equiv 3\unicode{x02007}(\,\text{mod 4}\,)\end{array}\right.$ and we also characterize the structure of all extremal trees with the maximum number of maximum generalized 4‐independent sets. A generalized k $k$‐independent set is a set of vertices such that the induced subgraph contains no trees with k $k$‐vertices, and the generalized k $k$‐independence number is the cardinality of a maximum k $k$‐independent set in G $G$. Zito proved that the maximum number of maximum generalized 2‐independent sets in a tree of order n $n$ is 2n−32 ${2}^{\frac{n-3}{2}}$ if n $n$ is odd, and 2n−22+1 ${2}^{\frac{n-2}{2}}+1$ if n $n$ is even. Tu et al. showed that the maximum number of maximum generalized 3‐independent sets in a tree of order n $n$ is 3n3−1+n3+1 ${3}^{\frac{n}{3}-1}+\frac{n}{3}+1$ if n≡0 (mod 3) $n\equiv 0\unicode{x02007}(\text{mod 3})$, and 3n−13−1+1 ${3}^{\frac{n-1}{3}-1}+1$ if n≡1 (mod 3) $n\equiv 1\unicode{x02007}(\text{mod 3})$, and 3n−23−1 ${3}^{\frac{n-2}{3}-1}$ if n≡2 (mod 3) $n\equiv 2\unicode{x02007}(\text{mod 3})$ and they characterized all the extremal graphs. Inspired by these two nice results, we establish four structure theorems about maximum generalized k $k$‐independent sets in a tree for a general integer k $k$. As applications, we show that the maximum number of generalized 4‐independent sets in a tree of order n (n≥4) $n\unicode{x02007}(n\ge 4)$ is 4n−44+(n4+1)(n8+1),n≡0 (mod 4),4n−54+n−14+1,n≡1 (mod 4),4n−64+n−24,n≡2 (mod 4),4n−74,n≡3 (mod 4) $\left\{\begin{array}{cc}{4}^{\frac{n-4}{4}}+(\frac{n}{4}+1)(\frac{n}{8}+1), & n\equiv 0\unicode{x02007}(\,\text{mod 4}\,),\\ {4}^{\frac{n-5}{4}}+\frac{n-1}{4}+1, & n\equiv 1\unicode{x02007}(\,\text{mod 4}\,),\\ {4}^{\frac{n-6}{4}}+\frac{n-2}{4}, & n\equiv 2\unicode{x02007}(\,\text{mod 4}\,),\\ {4}^{\frac{n-7}{4}}, & n\equiv 3\unicode{x02007}(\,\text{mod 4}\,)\end{array}\right.$ and we also characterize the structure of all extremal trees with the maximum number of maximum generalized 4‐independent sets. |
Author | Xu, Min Li, Pingshan |
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Cites_doi | 10.1002/jgt.20186 10.1016/j.disc.2012.09.010 10.1016/0012-365X(88)90114-8 10.1016/j.disc.2007.07.079 10.1016/j.dam.2011.04.023 10.3390/axioms1105019 10.1016/j.dam.2011.04.008 10.1002/jgt.20185 10.1002/jgt.22627 10.1137/0607015 10.1137/0401012 10.1016/S0166-218X(97)00033-4 10.1137/0210022 10.1016/j.tcs.2007.09.013 10.1002/jgt.3190110403 10.1007/BF02760024 10.1137/0406022 10.1007/s10107-005-0649-5 10.1002/jgt.3190150208 |
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Snippet | A generalized k $k$‐independent set is a set of vertices such that the induced subgraph contains no trees with k $k$‐vertices, and the generalized k... Abstract A generalized ‐independent set is a set of vertices such that the induced subgraph contains no trees with ‐vertices, and the generalized ‐independence... |
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SubjectTerms | Apexes Arrays dissociation sets Graph theory König–Egerváry graphs k‐independent sets tree Trees Trees (mathematics) |
Title | The maximum number of maximum generalized 4‐independent sets in trees |
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