CIIM 2016

Find all functions $f:(0,+\infty) \to (0,+\infty)$ that satisfy $(i)$ $f(xf(y))=yf(x), \forall x,y > 0,$ $(ii)$ $\lim_{x\to+\infty} f(x) = 0.$

A boa of size $k$ is a graph with $k+1$ vertices $\{0,1,\dots,k-1,k\}$ and edges only between the vertices $i$ and $i+1$ for $0\leq i < k.$ The boa is place in a graph $G$ through a injection of graphs. (This is an injective function form the vertices of the boa to the vertices of the graph in such a way that if there is an edge between the vertices $x$ and $y$ in the boa then there must be an edge between $f(x)$ and $f(y)$ in $G$ ). The Boa can move in the graph $G$ using to type of movement each time. If the boa is initially on the vertices $f(0),f(1),\dots,f(k)$ then it moves in one of the following ways:

(i) It choose $v$ a neighbor of $f(k)$ such that $v\not\in\{f(0),f(1),\dots,f(k-1)\}$ and the boa now moves to $f(0),f(1),\dots,f(k)$ with $f'(k)=v$ and $f'(i) = f(i+1)$ for $0 \leq i < k,$ or

(ii) It choose $v$ a neighbor of $f(0)$ such that $v\not\in\{f(1),f(2),\dots,f(k)\}$ and the boa now moves to $f(0),f(1),\dots,f(k)$ with $f'(0)=v$ and $f'(i) = f'(i-1)$ for $0 < i \leq k.$ Prove that if $G$ is a connected graph with diameter $d$ , then it is possible to put a size $\lceil d/2 \rceil$ boa in $G$ such that the boa can reach any vertex of $G$ .