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| Natura: | Preprint |
| Pubblicazione: |
2026
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| Accesso online: | https://arxiv.org/abs/2604.18769 |
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| _version_ | 1866910150130925568 |
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| author | Troyanov, Marc |
| author_facet | Troyanov, Marc |
| contents | We investigate the inner vertex-isoperimetric problem on the $d$-regular tree $T_d$. We first determine the exact value of the inner vertex-isoperimetric profile $I_d(k) = \min\{ |\partial D| \mid D\subset T_d \text{ finite and connected},\ |D|=k \}$, and we then introduce a boundary invariant, called the boundary branching excess $τ(D)$, and show that it provides a simple criterion for optimality. A domain $D\subset T_d$ is shown to be isoperimetrically optimal if and only if $τ(D)\le d-2$. Finally, we show that every domain in $T_d$ admits a canonical decomposition as an iterated gluing of full domains, namely domains whose entire boundary consists of leaves. This yields a complete description of all inner vertex-isoperimetric minimizers in $T_d$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_18769 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Isoperimetric Problem in Regular Trees Troyanov, Marc Combinatorics 05C05, 05C35, 05C10 We investigate the inner vertex-isoperimetric problem on the $d$-regular tree $T_d$. We first determine the exact value of the inner vertex-isoperimetric profile $I_d(k) = \min\{ |\partial D| \mid D\subset T_d \text{ finite and connected},\ |D|=k \}$, and we then introduce a boundary invariant, called the boundary branching excess $τ(D)$, and show that it provides a simple criterion for optimality. A domain $D\subset T_d$ is shown to be isoperimetrically optimal if and only if $τ(D)\le d-2$. Finally, we show that every domain in $T_d$ admits a canonical decomposition as an iterated gluing of full domains, namely domains whose entire boundary consists of leaves. This yields a complete description of all inner vertex-isoperimetric minimizers in $T_d$. |
| title | The Isoperimetric Problem in Regular Trees |
| topic | Combinatorics 05C05, 05C35, 05C10 |
| url | https://arxiv.org/abs/2604.18769 |