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The security of a network is closely related to the structure of the network graph. The denser the network graph structure is, the better it can resist attacks. Toughness and isolated toughness are used to characterize the vulnerable programs of the network which have been paid attention from mathematics and computer scholars. On this basis, considering the particularity of the sun component structures, sun toughness was introduced in mathematics and applied to computer networks. From the perspective of modern graph theory, this paper presents the sun toughness conditions of the path factor uniform graph and the path factor critical avoidable graph in  P ≥ 2 -factor and  P ≥ 3 -factor settings. Furthermore, examples show that the given boundaries are sharp.

journal of mathematics

Sun Toughness Conditions for <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <msub> <mrow> <mi>P</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </math> and <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"> <msub> <mrow> <mi>P</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msub> </math> Factor Uniform and Factor Critical Avoidable Graphs

Sun Toughness Conditions for P 2 and P 3 Factor Uniform and Factor Critical Avoidable Graphs

Sun Toughness Conditions for $P_{2}$ and $P_{3}$ Factor Uniform and Factor Critical Avoidable Graphs