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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2508.10077 |
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| _version_ | 1866909736564162560 |
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| author | Dankelmann, Peter Mafunda, Sonwabile Mallu, Sufiyan |
| author_facet | Dankelmann, Peter Mafunda, Sonwabile Mallu, Sufiyan |
| contents | Let $G$ be a finite, connected graph and $v$ a vertex of $G$. The average distance and the eccentricity of $v$ in $G$ are defined as the arithmetic mean and the maximum, respectively, of the distances from $v$ to all other vertices of $G$. The proximity of $G$ and the radius of $G$ are defined as the minimum of the average distances and the eccentricities over all vertices of $G$.
In this paper, we establish an upper bound on the proximity of a $2$-connected outerplanar graphs in terms of order and maximum face length. This bound is sharp apart from a small additive constant.
It is known that the radius of a maximal outerplanar graph is at most $\lfloor \frac{n}{4} \rfloor +1$. In the second part of this paper we show that this bound on the radius holds for a much larger subclass of outerplanar graphs, for all $2$-connected outerplanar graphs of order $n$ whose maximum face length does not exceed $\frac{n+2}{4}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_10077 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Proximity and Radius in Outerplanar Graphs with Bounded Faces Dankelmann, Peter Mafunda, Sonwabile Mallu, Sufiyan Combinatorics 05C12 Let $G$ be a finite, connected graph and $v$ a vertex of $G$. The average distance and the eccentricity of $v$ in $G$ are defined as the arithmetic mean and the maximum, respectively, of the distances from $v$ to all other vertices of $G$. The proximity of $G$ and the radius of $G$ are defined as the minimum of the average distances and the eccentricities over all vertices of $G$. In this paper, we establish an upper bound on the proximity of a $2$-connected outerplanar graphs in terms of order and maximum face length. This bound is sharp apart from a small additive constant. It is known that the radius of a maximal outerplanar graph is at most $\lfloor \frac{n}{4} \rfloor +1$. In the second part of this paper we show that this bound on the radius holds for a much larger subclass of outerplanar graphs, for all $2$-connected outerplanar graphs of order $n$ whose maximum face length does not exceed $\frac{n+2}{4}$. |
| title | Proximity and Radius in Outerplanar Graphs with Bounded Faces |
| topic | Combinatorics 05C12 |
| url | https://arxiv.org/abs/2508.10077 |