P<x>:=PolynomialRing(IntegerRing());
P;


//FOR THE DESCRIPTION OF ALL PROGRAMS, PLEASE SEE THE LAST ANNEXE OF OUR ARTICLE.

// Zeta Polynomial for loose trees

function PolTree(G)
		V:=Vertices(G);
		Q:=0;
		if #(V) eq 0 then return Q;
		else
			for i in [1..#(V)] do
				if Degree(V[i]) eq 1 then
					Q:=Q + 1;
				else
					Q:=Q + x^(Degree(V[i])) - x + 1;
				end if;
			end for;
		Q:=Q + x - 1;
		return Q;
		end if;		
end function;


//Given a loose graph G, create a spanning tree breaking a cycle in each step


function LooseSPTree(G)
		L:=[* G *];
		if IsTree(G) then
			return L;
		else
					
			G1:=SpanningTree(G);
			V1:=Vertices(G1);
			V:=Vertices(G);
			G2:=G;
			t:=#(V);
			t1:=#(V1);
			for i in [1 .. t] do
			A:=Neighbors(V[i]);
			A1:=Neighbors(V1[i]);
				for c in [i+1 .. t1] do
				if V[c] in A then
					if V1[c] notin A1 then
 						R:=G2 + 2;
						t2:=#(Vertices(R));
						R:=AddEdge(R, Vertices(R)[i], Vertices(R)[t2-1]);
						R:= R - { {Vertices(R)[i], Vertices(R)[c]} };
						G2:=AddEdge(R, Vertices(R)[c], Vertices(R)[t2]);
						L:=Append(L, G2);
					end if;
				end if;
				end for;
			end for;
			return L;
		end if;
end function;



//Given a graph, compute its Loose Spanning Tree

function PolLSPTree(G)
		B:=LooseSPTree(G);
		n:=#(B);
		A:=PolTree(B[n]);
		return A;
end function;


//Computing the graph G^L


function GL(A,B)
	V1:=Vertices(A);
	V2:=Vertices(B);
	t1:=#(V1);
	t2:=#(V2);
	L:=[* *];
	for i in [1 .. t2] do
		N2:=Neighbors(V2[i]);
		for c in [i+1 .. t2] do
			if V2[c] adj V2[i] then
				if V1[c] notadj V1[i] then
					H1:= Exclude(Neighbors(V2[c]), V2[i]);
					H2:= Exclude(Neighbors(V2[i]), V2[c]);
					H:= H1 meet H2;
					H3:= H1 sdiff H2;
					if IsEmpty(H) then
						L:=Append(L, NullGraph());
					else
						G3:=sub<B | H>;
						T:=Components(G3);
						lT:=#(T);
						for i in [1..lT] do
								G4:=sub<G3 | T[i]>;
								V4:=Vertices(G4);
								lV4:=#(V4);
								if #(H3) ge 1 then
									G5:=G4;
									VG5:=Vertices(G5);
									for j in [1..#(H3)] do
										V5:={@ v : v in VG5 | (Vertices(B) ! v) adj SetToIndexedSet(H3)[j] @};
										if IsEmpty(V5) then
										else
											G5:= G5 + 1;
											lG5:=#(Vertices(G5));
											if IsSubgraph(G5,G4) then
												V6:={@ (VertexSet(G5) ! V5[s]) :  s in [1..#(V5)] @};
												G5:= G5 + {{V6[s], Vertices(G5)[lG5]} : s in [1..#(V6)]};						
											end if;
										end if;
									end for;
									L:=Append(L,G5);
								else
									L:=[* sub<G3 | T[i]> : i in [1..lT] *];
								end if;								
						end for;
						
					end if;
					return L;
				end if;
			end if;
		end for;
	end for;
end function;	


//Computing the graph C(G^L, xy)

function GL2(A,B)
	V1:=Vertices(A);
	V2:=Vertices(B);
	t1:=#(V1);
	t2:=#(V2);
	for i in [1 .. t2] do
		N2:=Neighbors(V2[i]);
		for c in [i+1 .. t2] do
			if V2[c] adj V2[i] then
				if V1[c] notadj V1[i] then
					H1:= Exclude(Neighbors(V2[c]), V2[i]);
					H2:= Exclude(Neighbors(V2[i]), V2[c]);
					H:= H1 meet H2;
					H5:=Include(Include(H, V2[c]), V2[i]);
					H3:= H1 sdiff H2;
					G2:=sub<B | H5>;
					G3:=sub<B | H>;
					if IsEmpty(H) then
						L:=[* G2 *];
					else
						T:=Components(G3);
						lT:=#(T);
						G5:=G2;
						for i in [1..lT] do
								G4:=sub<G3 | T[i]>;
								V4:=Vertices(G4);
								lV4:=#(V4);
								if #(H3) ge 1 then
									for j in [1..#(H3)] do
										if IsSubgraph (G2,G4) then
										V5:={@ (VertexSet(B) ! (VertexSet(G3) ! v)) : v in V4 | (VertexSet(B) ! (VertexSet(G3) ! v)) adj SetToIndexedSet(H3)[j] @};
											if not IsEmpty(V5) then
												G5:=G5 + 1;
												lG5:=#(Vertices(G5));
												if IsSubgraph (G5,G2) then
													V6:={@ (VertexSet(G5) ! (VertexSet(G2) ! V5[s])) :  s in [1..#(V5)] @};
													G5:= G5 + {{V6[s], Vertices(G5)[lG5]} : s in [1..#(V6)]};	
												end if;			
											end if;
										end if;
									end for;	
								end if;	
						end for;
						L:=[* G5 *];
					end if;
					return L;
				end if;
			end if;
		end for;
	end for;
end function;

//Computing the graph G^L_x

function GLx(A,B)
	V1:=Vertices(A);
	V2:=Vertices(B);
	t1:=#(V1);
	t2:=#(V2);
	L:=[* *];
	for i in [1 .. t2] do
		N2:=Neighbors(V2[i]);
		for c in [i+1 .. t2] do
			if V2[c] adj V2[i] then
				if V1[c] notadj V1[i] then
					H1:= Exclude(Neighbors(V2[c]), V2[i]);
					H2:= Exclude(Neighbors(V2[i]), V2[c]);
					H:= H1 meet H2;
					H3:= H1 diff H2;
					if IsEmpty(H) then
						L:=Append(L, NullGraph());
					else
						G3:=sub<B | H>;
						T:=Components(G3);
						lT:=#(T);
						for i in [1..lT] do
								G4:=sub<G3 | T[i]>;
								V4:=Vertices(G4);
								lV4:=#(V4);
								if #(H3) ge 1 then
									G5:=G4;
									VG5:=Vertices(G5);
									for j in [1..#(H3)] do
										V5:={@ v : v in VG5 | (Vertices(B) ! v) adj SetToIndexedSet(H3)[j] @};
										if IsEmpty(V5) then
										else
											G5:= G5 + 1;
											lG5:=#(Vertices(G5));
											if IsSubgraph(G5,G4) then
												V6:={@ (VertexSet(G5) ! V5[s]) :  s in [1..#(V5)] @};
												G5:= G5 + {{V6[s], Vertices(G5)[lG5]} : s in [1..#(V6)]};						
											end if;
										end if;
									end for;
									L:=Append(L,G5);
								else
									L:=[* sub<G3 | T[i]> : i in [1..lT] *];
								end if;								
						end for;
						
					end if;
					return L;
				end if;
			end if;
		end for;
	end for;
end function;	


//Computing the graph C(G^L_x, xy)

function GL2x(A,B)
	V1:=Vertices(A);
	V2:=Vertices(B);
	t1:=#(V1);
	t2:=#(V2);
	for i in [1 .. t2] do
		N2:=Neighbors(V2[i]);
		for c in [i+1 .. t2] do
			if V2[c] adj V2[i] then
				if V1[c] notadj V1[i] then
					H1:= Exclude(Neighbors(V2[c]), V2[i]);
					H2:= Exclude(Neighbors(V2[i]), V2[c]);
					H:= H1 meet H2;
					H5:=Include(Include(H, V2[c]), V2[i]);
					H3:= H1 diff H2;
					G2:=sub<B | H5>;
					G3:=sub<B | H>;
					if IsEmpty(H) then
						L:=[* G2 *];
					else
						T:=Components(G3);
						lT:=#(T);
						G5:=G2;
						for i in [1..lT] do
								G4:=sub<G3 | T[i]>;
								V4:=Vertices(G4);
								lV4:=#(V4);
								if #(H3) ge 1 then
									for j in [1..#(H3)] do
										if IsSubgraph (G2,G4) then
										V5:={@ (VertexSet(B) ! (VertexSet(G3) ! v)) : v in V4 | (VertexSet(B) ! (VertexSet(G3) ! v)) adj SetToIndexedSet(H3)[j] @};
											if not IsEmpty(V5) then
												G5:=G5 + 1;
												lG5:=#(Vertices(G5));
												if IsSubgraph (G5,G2) then
													V6:={@ (VertexSet(G5) ! (VertexSet(G2) ! V5[s])) :  s in [1..#(V5)] @};
													G5:= G5 + {{V6[s], Vertices(G5)[lG5]} : s in [1..#(V6)]};	
												end if;			
											end if;
										end if;
									end for;	
								end if;	
						end for;
						L:=[* G5 *];
					end if;
					return L;
				end if;
			end if;
		end for;
	end for;
end function;


//Computing the graph C(G^L_x, y)


function GL3x(A,B)
	V1:=Vertices(A);
	V2:=Vertices(B);
	t1:=#(V1);
	t2:=#(V2);
	for i in [1 .. t2] do
		N2:=Neighbors(V2[i]);
		for c in [i+1 .. t2] do
			if V2[c] adj V2[i] then
				if V1[c] notadj V1[i] then
					H1:= Exclude(Neighbors(V2[c]), V2[i]);
					H2:= Exclude(Neighbors(V2[i]), V2[c]);
					H:= H1 meet H2;
					H5:=Include(H, V2[i]);
					H3:= H1 diff H2;
					G2:=sub<B | H5>;
					G3:=sub<B | H>;
					if IsEmpty(H) then
						L:=[* G2 *];
					else
						T:=Components(G3);
						lT:=#(T);
						G5:=G2;
						for i in [1..lT] do
								G4:=sub<G3 | T[i]>;
								V4:=Vertices(G4);
								lV4:=#(V4);
								if #(H3) ge 1 then
									for j in [1..#(H3)] do
										if IsSubgraph (G2,G4) then
										V5:={@ (VertexSet(B) ! (VertexSet(G3) ! v)) : v in V4 | (VertexSet(B) ! (VertexSet(G3) ! v)) adj SetToIndexedSet(H3)[j] @};
											if not IsEmpty(V5) then
												G5:=G5 + 1;
												lG5:=#(Vertices(G5));
												if IsSubgraph (G5,G2) then
													V6:={@ (VertexSet(G5) ! (VertexSet(G2) ! V5[s])) :  s in [1..#(V5)] @};
													G5:= G5 + {{V6[s], Vertices(G5)[lG5]} : s in [1..#(V6)]};	
												end if;			
											end if;
										end if;
									end for;	
								end if;	
						end for;
						L:=[* G5 *];
					end if;
					return L;
				end if;
			end if;
		end for;
	end for;
end function;
				

//Computing the graph G^L_y

function GLy(A,B)
	V1:=Vertices(A);
	V2:=Vertices(B);
	t1:=#(V1);
	t2:=#(V2);
	L:=[* *];
	for i in [1 .. t2] do
		N2:=Neighbors(V2[i]);
		for c in [i+1 .. t2] do
			if V2[c] adj V2[i] then
				if V1[c] notadj V1[i] then
					H1:= Exclude(Neighbors(V2[c]), V2[i]);
					H2:= Exclude(Neighbors(V2[i]), V2[c]);
					H:= H1 meet H2;
					H4:= H2 diff H1;
					if IsEmpty(H) then
						L:=Append(L, NullGraph());
					else
						G3:=sub<B | H>;
						T:=Components(G3);
						lT:=#(T);
						for i in [1..lT] do
								G4:=sub<G3 | T[i]>;
								V4:=Vertices(G4);
								lV4:=#(V4);
								if #(H4) ge 1 then
									G5:=G4;
									VG5:=Vertices(G5);
									for j in [1..#(H4)] do
										V5:={@ v : v in VG5 | (Vertices(B) ! v) adj SetToIndexedSet(H4)[j] @};
										if IsEmpty(V5) then
										else
											G5:= G5 + 1;
											lG5:=#(Vertices(G5));
											if IsSubgraph(G5,G4) then
												V6:={@ (VertexSet(G5) ! V5[s]) :  s in [1..#(V5)] @};
												G5:= G5 + {{V6[s], Vertices(G5)[lG5]} : s in [1..#(V6)]};						
											end if;
										end if;
									end for;
									L:=Append(L,G5);
								else
									L:=[* sub<G3 | T[i]> : i in [1..lT] *];
								end if;								
						end for;
						
					end if;
					return L;
				end if;
			end if;
		end for;
	end for;
end function;


//Computing the graph C(G^L_y, xy)

function GL2y(A,B)
	V1:=Vertices(A);
	V2:=Vertices(B);
	t1:=#(V1);
	t2:=#(V2);
	for i in [1 .. t2] do
		N2:=Neighbors(V2[i]);
		for c in [i+1 .. t2] do
			if V2[c] adj V2[i] then
				if V1[c] notadj V1[i] then
					H1:= Exclude(Neighbors(V2[c]), V2[i]);
					H2:= Exclude(Neighbors(V2[i]), V2[c]);
					H:= H1 meet H2;
					H5:=Include(Include(H, V2[c]), V2[i]);
					H3:= H2 diff H1;
					G2:=sub<B | H5>;
					G3:=sub<B | H>;
					if IsEmpty(H) then
						L:=[* G2 *];
					else
						T:=Components(G3);
						lT:=#(T);
						G5:=G2;
						for i in [1..lT] do
								G4:=sub<G3 | T[i]>;
								V4:=Vertices(G4);
								lV4:=#(V4);
								if #(H3) ge 1 then
									for j in [1..#(H3)] do
										if IsSubgraph (G2,G4) then
										V5:={@ (VertexSet(B) ! (VertexSet(G3) ! v)) : v in V4 | (VertexSet(B) ! (VertexSet(G3) ! v)) adj SetToIndexedSet(H3)[j] @};
											if not IsEmpty(V5) then
												G5:=G5 + 1;
												lG5:=#(Vertices(G5));
												if IsSubgraph (G5,G2) then
													V6:={@ (VertexSet(G5) ! (VertexSet(G2) ! V5[s])) :  s in [1..#(V5)] @};
													G5:= G5 + {{V6[s], Vertices(G5)[lG5]} : s in [1..#(V6)]};	
												end if;			
											end if;
										end if;
									end for;	
								end if;	
						end for;
						L:=[* G5 *];
					end if;
					return L;
				end if;
			end if;
		end for;
	end for;
end function;



//Computing the graph C(G^L_y, x)


function GL3y(A,B)
	V1:=Vertices(A);
	V2:=Vertices(B);
	t1:=#(V1);
	t2:=#(V2);
	for i in [1 .. t2] do
		N2:=Neighbors(V2[i]);
		for c in [i+1 .. t2] do
			if V2[c] adj V2[i] then
				if V1[c] notadj V1[i] then
					H1:= Exclude(Neighbors(V2[c]), V2[i]);
					H2:= Exclude(Neighbors(V2[i]), V2[c]);
					H:= H1 meet H2;
					H5:=Include(H, V2[c]);
					H3:= H2 diff H1;
					G2:=sub<B | H5>;
					G3:=sub<B | H>;
					if IsEmpty(H) then
						L:=[* G2 *];
					else
						T:=Components(G3);
						lT:=#(T);
						G5:=G2;
						for i in [1..lT] do
								G4:=sub<G3 | T[i]>;
								V4:=Vertices(G4);
								lV4:=#(V4);
								if #(H3) ge 1 then
									for j in [1..#(H3)] do
										if IsSubgraph (G2,G4) then
										V5:={@ (VertexSet(B) ! (VertexSet(G3) ! v)) : v in V4 | (VertexSet(B) ! (VertexSet(G3) ! v)) adj SetToIndexedSet(H3)[j] @};
											if not IsEmpty(V5) then
												G5:=G5 + 1;
												lG5:=#(Vertices(G5));
												if IsSubgraph (G5,G2) then
													V6:={@ (VertexSet(G5) ! (VertexSet(G2) ! V5[s])) :  s in [1..#(V5)] @};
													G5:= G5 + {{V6[s], Vertices(G5)[lG5]} : s in [1..#(V6)]};	
												end if;			
											end if;
										end if;
									end for;	
								end if;	
						end for;
						L:=[* G5 *];
					end if;
					return L;
				end if;
			end if;
		end for;
	end for;
end function;






//Create all list with all different graph from the whole surgery process


function ListGL (L)
	tL:=#(L);
	L1:=[* *];
	for i in [1..(tL-1)] do
		L1:=L1 cat GL(L[tL - i +1], L[tL - i]);
	end for;
	return L1;
end function;


function ListGLx (L)
	tL:=#(L);
	L1:=[* *];
	for i in [1..(tL-1)] do
		L1:=L1 cat GLx(L[tL - i +1], L[tL - i]);
	end for;
	return L1;
end function;


function ListGLy (L)
	tL:=#(L);
	L1:=[* *];
	for i in [1..(tL-1)] do
		L1:=L1 cat GLy(L[tL - i +1], L[tL - i]);
	end for;
	return L1;
end function;


function ListGL2 (L)
	tL:=#(L);
	L1:=[* *];
	for i in [1..(tL-1)] do
		L1:=L1 cat GL2(L[tL - i +1], L[tL - i]);
	end for;
	return L1;
end function;


function ListGL2x (L)
	tL:=#(L);
	L1:=[* *];
	for i in [1..(tL-1)] do
		L1:=L1 cat GL2x(L[tL - i +1], L[tL - i]);
	end for;
	return L1;
end function;


function ListGL2y (L)
	tL:=#(L);
	L1:=[* *];
	for i in [1..(tL-1)] do
		L1:=L1 cat GL2y(L[tL - i +1], L[tL - i]);
	end for;
	return L1;
end function;

function ListGL3x (L)
	tL:=#(L);
	L1:=[* *];
	for i in [1..(tL-1)] do
		L1:=L1 cat GL3x(L[tL - i +1], L[tL - i]);
	end for;
	return L1;
end function;


function ListGL3y (L)
	tL:=#(L);
	L1:=[* *];
	for i in [1..(tL-1)] do
		L1:=L1 cat GL3y(L[tL - i +1], L[tL - i]);
	end for;
	return L1;
end function;



//Keeping track of all vertices added in the surgery process


function mL (A,B)
	V1:=Vertices(A);
	V2:=Vertices(B);
	t1:=#(V1);
	t2:=#(V2);
	t:=0;
	for i in [1 .. t2] do
		N2:=Neighbors(V2[i]);
		for c in [i+1 .. t2] do
			if V2[c] adj V2[i] then
				if V1[c] notadj V1[i] then
					H1:= Exclude(Neighbors(V2[c]), V2[i]);
					H2:= Exclude(Neighbors(V2[i]), V2[c]);
					H:= H1 meet H2;
					L:=GL(A,B);
					for i in [1..#(L)] do
						t:= t + #(Vertices(L[i]));
					end for;
					t:= t - #(H);
				end if;
			end if;
		end for;
	end for;
	return t;
end function;



function ListmL(L)
	tL:=#(L);
	L2:=[mL(L[tL - i +1], L[tL - i]) : i in [1..(tL-1)]];
	return L2;
end function;

function mLx (A,B)
	V1:=Vertices(A);
	V2:=Vertices(B);
	t1:=#(V1);
	t2:=#(V2);
	t:=0;
	for i in [1 .. t2] do
		N2:=Neighbors(V2[i]);
		for c in [i+1 .. t2] do
			if V2[c] adj V2[i] then
				if V1[c] notadj V1[i] then
					H1:= Exclude(Neighbors(V2[c]), V2[i]);
					H2:= Exclude(Neighbors(V2[i]), V2[c]);
					H:= H1 meet H2;
					L:=GLx(A,B);
					for i in [1..#(L)] do
						t:= t + #(Vertices(L[i]));
					end for;
					t:= t - #(H);
				end if;
			end if;
		end for;
	end for;
	return t;
end function;



function ListmLx(L)
	tL:=#(L);
	L2:=[mLx(L[tL - i +1], L[tL - i]) : i in [1..(tL-1)]];
	return L2;
end function;


function mLy (A,B)
	V1:=Vertices(A);
	V2:=Vertices(B);
	t1:=#(V1);
	t2:=#(V2);
	t:=0;
	for i in [1 .. t2] do
		N2:=Neighbors(V2[i]);
		for c in [i+1 .. t2] do
			if V2[c] adj V2[i] then
				if V1[c] notadj V1[i] then
					H1:= Exclude(Neighbors(V2[c]), V2[i]);
					H2:= Exclude(Neighbors(V2[i]), V2[c]);
					H:= H1 meet H2;
					L:=GLy(A,B);
						for i in [1..#(L)] do
					t:= t + #(Vertices(L[i]));
					end for;
					t:= t - #(H);
				end if;
			end if;
		end for;
	end for;
	return t;
end function;



function ListmLy(L)
	tL:=#(L);
	L2:=[mLy(L[tL - i +1], L[tL - i]) : i in [1..(tL-1)]];
	return L2;
end function;


//Compute zeta polynomial for a graph


function PolynGraph1(G)
	Q:=0;
	if IsNull(G) then 
		return Q;
	else

		if IsTree(G) then
			return Q + PolTree(G);
		else
			V:=Vertices(G);
			D:=SetToIndexedSet(Alldeg(G, #(V) -1));
			if IsEmpty(D) then
				L:=LooseSPTree(G);
				tL:=#(L);
				Q:=Q + PolLSPTree(L[tL]) - 2*(tL - 1);
				L1:=ListGL(L);
				L2:=ListGLx(L);
				L3:=ListGLy(L);
				L4:=ListGL2(L);
				L5:=ListGL2x(L);
				L6:=ListGL2y(L);
				L7:=ListGL3x(L);
				L8:=ListGL3y(L);
				L9:=ListmL(L);
				L10:=ListmLx(L);
				L11:=ListmLy(L);
				for i in [1..#(L1)] do	
					Q:= Q - x^2*PolynGraph1(L1[i]);
				end for;
				for i in [1..#(L2)] do
					Q:= Q + (x-1)*PolynGraph1(L2[i]);
				end for;
				for i in [1..#(L3)] do
					Q:= Q + (x-1)*PolynGraph1(L3[i]);
				end for;
				for i in [1..#(L9)] do
					Q:= Q + (x^2-1)*L9[i];
				end for;
				for i in [1..#(L10)] do
					Q:= Q - (x-1)*L10[i];
				end for;
				for i in [1..#(L11)] do
					Q:= Q - (x-1)*L11[i];
				end for;
				for i in [1..#(L4)] do
					Q:= Q  + PolynGraph1(L4[i]);
				end for;
				for i in [1..#(L5)] do
					Q:= Q  - PolynGraph1(L5[i]);
				end for;
				for i in [1..#(L6)] do
					Q:= Q  - PolynGraph1(L6[i]);
				end for;
				for i in [1..#(L7)] do
					Q:= Q  + PolynGraph1(L7[i]);
				end for;
				for i in [1..#(L8)] do	
					Q:= Q  + PolynGraph1(L8[i]);
				end for;
			else 
				G1:= G - D[1];
				G2:= Components(G1);
				Q:=Q + x^(Degree(D[1])); 
				for i in [1..#(G2)] do
						Q:= Q + PolynGraph1(sub<G1 | G2[i]>);
				end for;
			end if;
			return Q;
		end if;
	end if;
end function;

//Compute the zeta polynomial for a loose graph

function PolynGraph(G,m)
	Q:= PolynGraph1(G) - m;
	return Q;
end function;


//EXAMPLES OF GRAPHS

A:= Graph< 6 | {1,2}, {1,3}, {1,4}, {1,5}, {2,3}, {3,4}, {3,5}, {3,6}, {4,5}, {5,6}>;

B:=Graph< 5 | {1,2}, {2,3}, {2,4}, {3,4}, {4,5}>;

C:=Graph<5 | {1,2}, {1,3}, {1,4}, {1,5}, {2,3}, {2,4}, {2,5}, {3,4}, {3,5}, {4,5}>;

D:= Graph< 7 | {1,2}, {1,3}, {1,4}, {1,5}, {2,3}, {3,4}, {3,5}, {3,6}, {4,5}, {5,6}, {1,7}, {7,3}>;

E:=Graph <5 | {5,2}, {5,3}, {5,4}, {1,5}, {2,3}, {3,4}, {4,1}, {1,2}>;

F:= Graph <6 | {1,2}, {1,3}, {1,4}, {1,5}, {2,3}, {3,4}, {4,5}, {5,2}, {6,2} >;
 
G:=Graph<6 | {1,2}, {1,3}, {1,4}, {1,5}, {2,3}, {3,4}, {4,5}, {5,2}, {6,2}, {6,3}, {6,4}, {6,5}>;   
                                         
H:=Graph<7 | {1,2}, {1,3}, {1,4}, {1,5}, {1,6}, {2,5}, {2,6}, {2,7}, {3,6}, {3,5}, {4,5}, {6,7}>;

I:=Graph<7 | {1,2}, {1,3}, {1,4}, {1,5}, {1,6}, {2,5}, {2,6}, {2,7}, {3,6}, {3,5}, {4,5}, {6,7}, {5,6}>;

J:=Graph<8 | {1,2}, {1,3}, {1,8}, {2,6}, {2,7}, {3,4}, {3,6}, {4,5}, {4,8}, {5,6}, {5,7}, {7,8}>;