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Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula> be a class of given graphs. A graph <i>G</i> is said to be <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula>-free if <i>G</i> contains no induced copies of <i>H</i> for any <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>∈</mo><mi mathvariant="script">H</mi></mrow></semantics></math></inline-formula>. In this article, we characterize all connected subgraph pairs <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><mi>R</mi><mo>,</mo><mi>S</mi><mo>}</mo></mrow></semantics></math></inline-formula> guranteeing the edge-connectivity of a connected <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><mi>R</mi><mo>,</mo><mi>S</mi><mo>}</mo></mrow></semantics></math></inline-formula>-free graph to have the same minimum degree. Our result is a supplement of Wang et al. Furthermore, we obtain a relationship of forbidden sets when those general parameters have the recurrence relation.