The edge surviving rate of Halin graphs

<p>Let <span class="math inline">\(k\ge 1\)</span> be an integer. Let <span class="math inline">\(G=(V,E)\)</span> be a connected graph with <span class="math inline">\(n\)</span> vertices and <span class="math inline">\(m\)</span> edges. Suppose fires break out at two adjacent vertices. In each round, a firefighter can protect <span class="math inline">\(k\)</span> vertices, and then the fires spread to all unprotected neighbors. For <span class="math inline">\(uv\in E(G)\)</span>, let <span class="math inline">\(sn_{k}(uv)\)</span> denote the maximum number of vertices the firefighter can save when fires break out at the ends of <span class="math inline">\(uv\)</span>. The <span class="math inline">\(k\)</span>-edge surviving rate <span class="math inline">\(\rho&#39;_k(G)\)</span> of <span class="math inline">\(G\)</span> is defined as the average proportion of vertices saved when the starting vertices of the fires are chosen uniformly at random over all eages, i.e., <span class="math inline">\(\rho&#39;_k(G)=\sum\limits_{uv\in E(G)}sn_{k}(uv)/nm\)</span>. In particular, we write <span class="math inline">\(\rho&#39;(G)=\rho&#39;_1(G)\)</span>. For a given class of graphs <span class="math inline">\(\mathcal{G}\)</span> and a constant <span class="math inline">\(\varepsilon>0\)</span>, we seek the minimum value <span class="math inline">\(k\)</span> such that <span class="math inline">\(\rho&#39;_k(G)>\varepsilon\)</span> for all <span class="math inline">\(G\in \mathcal{G}\)</span>. In this paper, we prove that for Halin graphs, this minimum value is exactly 1. Specifically, every Halin graph <span class="math inline">\(G\)</span> satisfies <span class="math inline">\(\rho&#39;(G)> 1/12\)</span>.</p>