The maximum number of maximum generalized 4‐independent sets in trees

• Published in: Journal of graph theory Vol. 107; no. 2; pp. 359 - 380
• Main Authors: Li, Pingshan, Xu, Min
• Format: Journal Article
• Language: English
• Published: Hoboken Wiley Subscription Services, Inc 01.10.2024
• Subjects: Apexes; Arrays; dissociation sets; Graph theory; König–Egerváry graphs; k‐independent sets; tree; Trees; Trees (mathematics)
• ISSN: 0364-9024; 1097-0118
• DOI: 10.1002/jgt.23122

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.