v\in G(I)$.

By (\[lcm\]) the ideal $I$ is linearly related if and only if all the open intervals $(1,w)$ with $w\in L(I)$ and $\deg w>d+1$ are connected. Considering the Taylor complex of $I$ we see that $\beta_{1,w}(I)=0$ if there exists no $u ,v \in G(I)$ such that $w=\lcm(u,v)$. Thus we need only to consider intervals $(1,w)$ with $w=\lcm(u,v)$ for some $u ,v \in G(I)$, and hence $I$ is linearly related if and only if $\Delta((1,\lcm(u, v)))$ is connected for all $u,v\in G(I)$ with $\deg(\lcm(u,v))>d+1$.

We first assume that $G_I^{(u,v)}$ is connected for all $u,v \in G(I)$. Now let $u,v \in G(I)$ with $\deg(\lcm(u,v))>d+1$. Let $C$ and $C'$ be maximal chains of the interval $(1,\lcm(u,v))$ (i.e.  facets of $\Delta((1,\lcm(u,v)))$). For a chain $D$ in $(1,\lcm(u,v))$ we denote by $\min(D)$ the minimal element in $D$. Obviously, $\min(D)\in V(G_I^{(u,v)})$. Let $w=\min(C)$ and $w'=\min(C')$. Then $w,w'\in V(G^{(u,v)}_I)$. Hence there exists a sequence $w_{1},\ldots,w_{r}\in V(G^{(u,v)}_I)$ with $w=w_{1}$ and $w'=w_{r}$ and such that the degree of $v_j:=\lcm(w_{j},w_{j+1})$ is $d+1$ for $j=1,\ldots,r-1$. Since $v_j$ divides $\lcm(u,v)$ and since $\deg v_j<\deg (\lcm(u,v))$ it follows that $v_j\in (1,\lcm(u,v))$. Thus there exist maximal chains $C_j$ and $D_j$ with $w_{j},v_j\in C_j$ and $v_j,w_{j+1}\in D_j$. Consider the sequence of maximal chains $$C,C_1,D_1,C_2,D_2,\ldots,C_{r-1},D_{r-1}, 