in $G(I)$ ordered by divisibility. The unique minimal element in $L(I)$ is $1$. For any $u\in L(I)$ we denote by $(1,u)$ the open interval of $L(I)$ which by definition is the induced subposet of $L(I)$ with elements $v\in L(I)$ with $1<v<u$. Furthermore, we denote by $\Delta((1,u))$ the order complex of the poset $(1,u)$.

The minimal graded free resolution of $I$ is multi-graded. Identifying a monomial with its multi-degree we denote the multi-graded Betti numbers of $I$ by $\beta_{i,u}(I)$ where $i$ is the homological degree and $u$ is a monomial. By Gasharov, Peeva and Welker one has $$\begin{aligned}
\label{lcm}
\beta_{i,u}(I)=\dim_K \widetilde{H}_{i-1}(\Delta((1,u)); K) \quad \text{for all $i\geq 0$ and all $u\in L(I)$.}\end{aligned}$$ Moreover, $\beta_{i,u}(I)=0$ if $u\not \in L(I)$.

Now suppose that all generators of $I$ are of degree $d$. We define the graph $G_I$ whose vertex set is $G(I)$ and for which $\{u,v\}$ is an edge of $G_I$ if and only if $\deg (\lcm(u,v))=d+1$.

For all $u, v\in G(I)$ let $G^{(u,v)}_I$ be the induced subgraph of $G_I$ with vertex set $$V(G^{(u,v)}_I)=\{w\: \text{$w$ divides $\lcm(u, v)$}\}.$$

\[graph\] \[criterion\] Let $I$ be a monomial ideal generated in degree $d$. Then $I$ is linearly related if and only if $G^{(u,v)}_I$ is connected for all $u, 