On critically $3$-connected graphs with exactly two vertices of degree 3. Part 1

A graph $G$ is critically $3$-connected, if $G$ is $3$-connected and for any vertex $v\in V(G)$ the graph $G-v$ isn't $3$-connected. R. C. Entringer and P. J. Slater proved that any critically $3$-connected graph contains at least two vertices of degree 3. In this paper we classify all such graphs with one additional condition: two vertices of degree 3 are adjacent. The case of nonadjacent vertices of degree 3 will be investigated in the second part of the paper, which will be published later.