On Burgess's theorem and related problems

Let $G$ be a graph. We determine all graphs which are $G$-like. We also prove that if $G_{i} (i=1,2,\ldots , m)$ are graphs, then in order that each $G_{i}$-like $(i=1,2,\ldots , m)$ continuum $M$ be $n$-indecomposable for some $n=n(M)$ it is necessary and sufficient that if $K$ is a graph, then $K$ is not $G_{i}$-like for some integer $i$ with $1\le i\le m$. This generalizes a well known theorem of Burgess.