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We provide explicit graded constructions of orbifold del Pezzo surfaces with rigid orbifold points of type <inline-formula><math display="inline"><semantics><mfenced separators="" open="{" close="}"><msub><mi>k</mi><mi>i</mi></msub><mo>×</mo><mfrac><mn>1</mn><msub><mi>r</mi><mi>i</mi></msub></mfrac><mrow><mo>(</mo><mn>1</mn><mo>,</mo><msub><mi>a</mi><mi>i</mi></msub><mo>)</mo></mrow><mo>:</mo><mn>3</mn><mo>≤</mo><msub><mi>r</mi><mi>i</mi></msub><mo>≤</mo><mn>10</mn><mo>,</mo><msub><mi>k</mi><mi>i</mi></msub><mo>∈</mo><msub><mi mathvariant="double-struck">Z</mi><mrow><mo>≥</mo><mn>0</mn></mrow></msub></mfenced></semantics></math></inline-formula> as well-formed and quasismooth varieties embedded in some weighted projective space. In particular, we present a collection of 147 such surfaces such that their image under their anti-canonical embeddings can be described by using one of the following sets of equations: a single equation, two linearly independent equations, five maximal Pfaffians of <inline-formula><math display="inline"><semantics><mrow><mn>5</mn><mo>×</mo><mn>5</mn></mrow></semantics></math></inline-formula> skew symmetric matrix, and nine <inline-formula><math display="inline"><semantics><mrow><mn>2</mn><mo>×</mo><mn>2</mn></mrow></semantics></math></inline-formula> minors of size 3 square matrix. This is a complete classification of such surfaces under certain carefully chosen bounds on the weights of ambient weighted projective spaces and it is largely based on detailed computer-assisted searches by using the computer algebra system MAGMA.