Game extensions of floppy graph metrics

Abstract: A $graph$ $metric$ on a set $X$ is any function $d: E_d \to\mathbb R_+:={x\in\mathbb R:x>0}$ defined on a connected graph $ E_d \subseteq[X]^2:={A\subseteq X:|A|=2}$ and such that for every ${x,y}\in E_d$ we have $d({x,y})\le\hat d(x,y):=\inf\big{\sum_{i=1}^nd({x_{i-1},x_i}):{x,y}={x_0,x_n}\;\wedge\;{{x_{i-1},x_i}:0<i\le n}\subseteq E_d \big}$. A graph metric $d$ is called a $full$ $metric$ on $X$ if $ E_d =[X]^2$. A graph metric $d: E_d \to\bar{\mathbb R}+$ is $floppy$ if $\hat d(x,y)>\check d(x,y:= \sup{d({a,b})-\hat d(a,u)-\hat d(b,y):{a,b}\in E_d }$ for every $x,y\in X$ with ${x,y}\notin E_d $. We prove that for every floppy graph metric $d: E_d \to\mathbb R+$ on a set $X$, every points $x,y\in X$ with ${x,y}\notin E_d $, and every real number $r$ with $\frac 13\check d(x,y)+\frac23\hat d(x,y)\le r<\hat d(x,y)$ the function $d\cup{\langle{x,y},r\rangle}$ is a floppy graph metric. This implies that for every floppy graph metric $d: E_d \to\mathbb R_+$ with countable set $[X]^2\setminus E_d $ and for every indexed family $(F_e){e\in[X]^2\setminus E_d }$ of dense subsets of $\mathbb R+$, there exists an injective function $r\in\prod_{e\in[X]^2\setminus E_d}F_e$ such that $d\cup r$ is a full metric. Also, we prove that the latter result does not extend to partial metrics defined on uncountable sets.