A graph G is called to be a 2-degree integral subgraph of a q-tree if it is obtained by deleting an edge e from an integral subgraph that is contained in exactly q - 1 triangles. An added-vertex q-tree G with n vertices is obtained by taking two vertices u, v (u, v are not adjacent) in a q-trees T with n - 1 vertices such that their intersection of neighborhoods of u, v forms a complete graph $K_{q}$, and adding a new vertex x, new edges xu, xv, $xv_{1},;xv_{2},;{cdots},;xv_{q-4}$, where ${v_{1},;v_{2},;{cdots},;v_{q-4}};{subseteq};K_{q}$. In this paper we prove that a graph G with minimum degree not equal to q - 3 and chromatic polynomial $$P(G;{lambda});=;{lambda}({lambda}-1);{cdots};({lambda}-q+2)({lambda}-q+1)^{3}({lambda}-q)^{n-q-2}$$ with $n;{geq};q+2$ has and only has 2-degree integral subgraph of q-tree with n vertices and added-vertex q-tree with n vertices.