Needs["DiscreteMath`Combinatorica`"];

ToleranceLG = 10^(-9);

(* ProxGRaphDisk in 1 dimension *)
ProxGraphDisk1D[pointset_,radius_]:= Module[{vertices=pointset,edges,n=Length[pointset],ii,tt,kk},
   edgesCheck = Flatten[NonPositive[Table[Table[If[Abs[ pointset[[tt]]- pointset[[ii]]] < ToleranceLG,1,Abs[ pointset[[tt]]- pointset[[ii]]] - radius - ToleranceLG],{ii,tt+1,n}],{tt,1,n-1}]]];
   edges = Delete[KSubsets[Table[kk,{kk,1,n}],2],Position[Flatten[edgesCheck],False]];
   Return[{edges,vertices}];
];

ProxGraphComplete1D[pointset_]:= {KSubsets[Table[kk,{kk,1,Length[pointset]}],2],pointset};

ToAdjacencyLists1D[graph1D_]:= Module[{NumberVertices=Length[graph1D[[2]]],ii},
   edges=Partition[graph1D[[1]],1];
   vertices = Partition[Table[{0,0},{ii,1,NumberVertices}],1];
   AdjacencyLists1D=ToAdjacencyLists[Graph[edges,vertices]];
   Return[AdjacencyLists1D];
];