Summary: The secure-domination number \(\gamma_s(G)\) of a finite simple graph \(G\) is the smallest size of a set of vertices in \(G\) which is both secure and dominating in \(G\). It is elementary that \(\frac {\gamma_s(G)}{|V(G)|}\geq\frac 12\). A currently unsolved problem:
Find the largest number \(g\in(\frac 12,1)\) such that there is a sequence \((H_n)\) of distinct connected graphs such that \[ \lim_{n\to\infty}\frac {\gamma_s(H_n)}{|V(H_n)|}=g. \]
In early work on this problem the full balanced binary trees \(G_n\) (which has \(2^n\) leafs, each a distance \(n\) from the root vertex) were of interest. They are no longer, but the unexpected difficulty of determining \(\gamma_s(G_n)\) has made this determination of interest in itself. In this paper we show that \[ \lim\inf_{n\to\infty}\frac{\gamma_s(G_n)}{|V(G_n)|}\geq\frac{8}{15} \] and give a construction of a secure-dominating set \(S_n\subseteq V(G_n)\) such that \[ \lim_{n\to\infty}\frac{|S_n|}{V(G_n)}= \frac{11}{20}. \] However, we find that \(S_n\) is not a minimum secure-dominating set in \(G_n\) for \(n\geq 6\).