The Outerplanar Tur\'{a}n Number of Double Stars

TL;DR

This paper determines the maximum number of edges in outerplanar graphs that do not contain a specific double star as a subgraph, providing exact values for most cases.

Contribution

It offers the first comprehensive determination of the outerplanar Turán number for double stars, with a notable exception and a new lower bound.

Findings

Exact values of $ex_{_ ext{OP}}(n,S_{p,q})$ for all $q extgreater p extgreater 1$, except $p=2$, $q=3$

Established a lower bound for the case $p=2$, $q=3$

Extended understanding of extremal outerplanar graphs avoiding double stars.

Abstract

Let $H$ be a nonempty graph. A graph is $H$-free if it does not contain any copy of $H$ as a subgraph. The outerplanar Tur\'{a}n number of $H$, denoted by $ex_{_\mathcal{OP}}(n,H)$, is the maximum number of edges among all $H$-free outerplanar graphs on $n$ vertices. A double star $S_{p,q}$ is the graph obtained from an edge by joining its two endpoints with $p$ and $q$ isolated vertices respectively, where $q \ge p\ge 1$. In this paper, we determine the exact values of $ex_{_\mathcal{OP}}(n,S_{p,q})$ for all $q\ge p\ge 2$, with the sole exception of $p=2$ and $q=3$; for the latter, we establish a lower bound.