Graph[{1, 2, 3, 4}, {UndirectedEdge[1, 2], UndirectedEdge[1, 3], UndirectedEdge[1, 4],
UndirectedEdge[2, 4], UndirectedEdge[2, 3], UndirectedEdge[3, 4]},
{EdgeStyle -> {Directive[GrayLevel[0.5], Thickness[Large]]},
GraphLayout -> {Dimension -> 2}, PerformanceGoal -> Quality,
VertexCoordinates -> {{0.880095814601738, 0.013156064124647815},
{0.013242690904806398, 0.}, {0.8663279571038569, 0.8780079864151075}, {0.,
0.8656690492208879}}, VertexShapeFunction ->
{Function[{center$, name$, size$},
If[name$ <= Length[{{6, 9, 12}, {11, 7, 1}, {5, 10, 2}, {8, 4, 3}}],
IconObject[center$, (If[IntegerQ[#1], {coordinates$7149[[Quotient[2 + #1, 3]]],
#1, name$}, {coordinates$7149[[Length[{{6, 9, 12}, {11, 7, 1}, {5, 10, 2}, {
8, 4, 3}}] + First[#1]]], #1}] & ) /@
{{6, 9, 12}, {11, 7, 1}, {5, 10, 2}, {8, 4, 3}}[[name$]], Max[size$]],
{LightGray, Dotted, Circle[center$, Max[size$]]}]]}, VertexSize -> {Small},
VertexStyle -> {Directive[GrayLevel[0.5], Thickness[Large]]}}]