Let　G　be　a　graph　which　may　have　multiple　edges　but　no　loops,　μG(v)　be　the　multiplicity　of　G　(which　is　the　maximum　among　the　numbers　of　edges　between　v　and　the　other　vertex　on　G),　and　a　and　b　be　any　two　functions,　s.t.　a,b:V(G)→N\{0,1}.　A　theta　is　a　graph　made　of　three　internally　vertex-disjoint　chordless　paths　P1=x-⋯-y,　P2=x-⋯-y,　P3=x-⋯-y　of　length　at　least　two　and　no　edges　exist　between　Pi　and　Pj　(i≠j,i,j∈{1,2,3})　except　the　three　edges　incident　to　x　and　the　three　edges　incident　to　y.　For　a　partition　(X, Y)　of　V(G),　and　any　x∈X,　y∈Y,　let　dX(x)　denote　by　the　degree　of　x　in　G[X],　and　dY(y)　be　defined　similarly.　In　this　paper,　we　show　that　a　graph　G　admits　a　partition　(A, B)　such　that　dA(x)≥a(x)　for　any　x∈A　and　dB(y)≥b(y)　for　any　y∈B　if　G　is　H-free　and　dG(v)≥a+b+2μG(v)-3　for　any　vertex　v∈V(G),　where　for　each　member　H∈H,　the　underlying　of　H　belongs　to　{triangle,　wheel,　theta}.　©　The　Author(s)　under　exclusive　licence　to　Sociedade　Brasileira　de　Matemática　Aplicada　e　Computacional　2024.