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- W2000223848 abstract "The bridge degree<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$mathrm{bdeg}v$><mml:mrow><mml:mtext>bdeg</mml:mtext><mml:mi> </mml:mi><mml:mi>v</mml:mi></mml:mrow></mml:math>and cycle degree<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$mathrm{cdeg}v$><mml:mrow><mml:mtext>cdeg</mml:mtext><mml:mi> </mml:mi><mml:mi>v</mml:mi></mml:mrow></mml:math>of a vertex<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$v$><mml:mi>v</mml:mi></mml:math>in a graph<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$G$><mml:mi>G</mml:mi></mml:math>are, respectively, the number of bridges and number of cycle edges incident with<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$v$><mml:mi>v</mml:mi></mml:math>in<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$G$><mml:mi>G</mml:mi></mml:math>. A characterization of finite nonempty sets<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$S$><mml:mi>S</mml:mi></mml:math>of nonnegative integers is given for which<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$S$><mml:mi>S</mml:mi></mml:math>is the set of bridge degrees (cycle degrees) of the vertices of some graph. The bridge-cycle degree of a vertex<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$v$><mml:mi>v</mml:mi></mml:math>in a graph<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$G$><mml:mi>G</mml:mi></mml:math>is the ordered pair<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$left( {b,c} right)$><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>b</mml:mi><mml:mo>,</mml:mo><mml:mi>c</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>, where<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$mathrm{bdeg}v = b$><mml:mrow><mml:mtext>bdeg</mml:mtext><mml:mi> </mml:mi><mml:mi>v</mml:mi><mml:mo>=</mml:mo><mml:mi>b</mml:mi></mml:mrow></mml:math>and<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$mathrm{cdeg}v = c$><mml:mrow><mml:mtext>cdeg</mml:mtext><mml:mi> </mml:mi><mml:mi>v</mml:mi><mml:mo>=</mml:mo><mml:mi>c</mml:mi></mml:mrow></mml:math>. Those finite sets<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$S$><mml:mi>S</mml:mi></mml:math>of ordered pairs of nonnegative integers for which<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$S$><mml:mi>S</mml:mi></mml:math>is the set of bridge-cycle degrees of the vertices of some graph are also characterized." @default.
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- W2000223848 date "1984-01-01" @default.
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- W2000223848 title "Bridge and cycle degrees of vertices of graphs" @default.
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