Broadcast domination models the idea of covering a network of cities by transmitters of varying powers while minimizing the total cost of the transmitters used to achieve full coverage. To be exact, let G be a connected graph of order at least two with vertex set $V(G)$ and edge set $E(G)$. Let $d(x,y)$, $e(v)$, and $\mathrm{diam}(G)$, respectively, denote the length of a shortest $x-y$ path in G, the eccentricity of a vertex v in G, and the diameter of G. A function $f:V(G)\to \{0,1,\dots ,\mathrm{diam}(G)\}$ is called a broadcast if $f(v)\le e(v)$ for each $v\in V(G)$. A broadcast f is called a dominating broadcast of G if, for each vertex $u\in V(G)$, there exists a vertex $v\in V(G)$ such that $f(v)>0$ and $d(u,v)\le f(v)$. The broadcast domination number, ${\gamma }_b(G)$, of G is the minimum of ${\sum }_{v\in V(G)}f(v)$ over all dominating broadcasts f on G. Let $G_1$ and $G_2$ be two disjoint copies of a graph G, and let $\sigma :V(G_1)\to V(G_2)$ be a bijection. Then a permutation graph $G_{\sigma }=(V,E)$ has vertex set $V=V(G_1)\cup V(G_2)$ and edge set $E=E(G_1)\cup E(G_2)\cup \{uv:v=\sigma (u)\}$. For a connected graph G of order at least two, we prove the sharp bounds $2\le {\gamma }_b(G_{\sigma })\le 2{\gamma }_b(G)$; we give an example showing that there is no function h such that ${\gamma }_b(G)<h({\gamma }_b(G_{\sigma }))$ for all pairs $(G,\sigma )$. We characterize $G_{\sigma }$ satisfying ${\gamma }_b(G_{\sigma })=2$, and examine ${\gamma }_b(G_{\sigma })$ when G is a cycle, a path, or a complete multipartite graph.

广播统治模式的构想是，用功率不同的发射机覆盖城市网络，同时将用于实现全覆盖的发射机的总成本降至最低。确切地说，令G为顶点集至少为2的阶连通图$V（G）$ 和边缘集 $Ë（G）$。让$d（X，ÿ）$， $Ë（v）$和 $\mathrm{直径}（G）$分别表示最短的长度 $X-ÿ$在路径G ^，顶点的偏心v在ģ，和直径ģ。功能$F：V（G）\to \{0，\mathrm{1个}，\dots ，\mathrm{直径}（G）\}$被称为广播，如果$F（v）\le Ë（v）$ 每个 $v\in V（G）$。广播˚F称为主导广播的ģ如果，为每个顶点$ü\in V（G）$，存在一个顶点 $v\in V（G）$ 这样 $F（v）>0$ 和 $d（ü，v）\le F（v）$。该广播控制数，${\gamma }_b（G）$中的，是G的最小值${\sum }_{v\in V（G）}F（v）$在所有主导广播˚F上摹。让$G_{\mathrm{1个}}$ 和 $G_2$是图G的两个不相交的副本，并令$\sigma ：V（G_{\mathrm{1个}}）\to V（G_2）$是一个双射。然后是一个置换图 $G_{\sigma }=（V，Ë）$ 有顶点集 $V=V（G_{\mathrm{1个}}）\cup V（G_2）$ 和边缘集 $Ë=Ë（G_{\mathrm{1个}}）\cup Ë（G_2）\cup \{üv：v=\sigma （ü）\}$。对于阶数至少为2的连通图G，我们证明了鲜明的界限$2\le {\gamma }_b（G_{\sigma }）\le 2{\gamma }_b（G）$; 我们举一个例子，说明没有函数h使得${\gamma }_b（G）<H（{\gamma }_b（G_{\sigma }））$ 对于所有对 $（G，\sigma ）$。我们表征$G_{\sigma }$ 满意的 ${\gamma }_b（G_{\sigma }）=2$，然后检查 ${\gamma }_b（G_{\sigma }）$当G是循环，路径或完整的多部分图时。