Toughness condition for a graph to be all fractional (g,f,n)-critical deleted

Gao, Wei; Wang, Weifan; Dimitrov, Darko

doi:10.2298/fil1909735g

Cited by 24 publications

(7 citation statements)

References 32 publications

(43 reference statements)

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“…Since ≥ a 1, we confirm that there is only the following possible combination of a b c ( , , ): (1,0,0), (2, 0, 0), (3, 0, 0), (1,1,0). By (11), the only possibility is a b c ( , , ) = (3, 0, 0). In this case, deleting one vertex as S in G′ and resulting subgraph K…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Enomoto et al 6 even introduced a variant of toughness, denoted by

τ (G)

and stated as follows:

τ (G) = + \infty

G

is a complete graph; and

τ (G) = \min (}{, \frac{∣ S ∣}{ω (G - S) - 1} : ω (G - S) \geq 2)

for noncomplete graph. Recently, Gao et al 10,11 continue working on this topic, and several toughness bounds are obtained to ensure a graph has fractional factors in different settings. On the other hand, the feasibility of network data transmission is equivalent to the existence of fractional flow in the network, and after modeling with graphs, it can be characterized by the existence of factors or fractional factors in various settings.…”

Section: Introductionmentioning

confidence: 99%

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Tight bounds for the existence of path factors in network vulnerability parameter settings

2020

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The issues of ruggedness and vulnerability are cruxes in network security research, which must be considered during the network designing phase. Parameters such as toughness, isolated toughness, and binding number characterize the vulnerable of the network from the structure of networks. The path factor, a special case of the generalized MJX-tex-caligraphicnormalℋ‐factor, measures the feasibility of data transmission in networks. Recent advances have been obtained to show that there is an inevitable connection between the vulnerability parameters of the network and the existence of path factors, while we found that some existing theoretical results are not tight and there is still a long way for further improvement. In view of graph theory approaches, this paper mainly contributes to determine the sharp bounds of toughness, isolated toughness, and binding number for the existence of path factor in different settings, and therefore solve the open problems left unsolved in previous articles.

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Section: Proof Of Theoremmentioning

confidence: 99%

“…Enomoto et al 6 even introduced a variant of toughness, denoted by

τ (G)

and stated as follows:

τ (G) = + \infty

G

is a complete graph; and

τ (G) = \min (}{, \frac{∣ S ∣}{ω (G - S) - 1} : ω (G - S) \geq 2)

Section: Introductionmentioning

confidence: 99%

Tight bounds for the existence of path factors in network vulnerability parameter settings

2020

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“…After decades of studying, scholars have found that these parameters are inherently related to the existence of fractional factors (see References [9][10][11]), and the latter characterizes the feasibility of information transmission in the network. After that, the researchers discovered that these parameters are closely related to the existence of some special -factors (see References [12][13][14]).…”

Section: Introductionmentioning

confidence: 99%

Network vulnerability parameter and results on two surfaces

2021

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Isolation toughness is a vital parameter to evaluate the vulnerability of computer networks. In specific network designing stage, it is necessary to find the lower bound of the isolated toughness, and strive to build a network that meets the stability requirements with the least cost. Gao et al.1 conjectured that if a graph G with κ ( G ) ≥ 3 m + 1 2 satisfies I ( G ) > 7 m + 5 4 m + 4 or I ′ ( G ) > 7 m + 5 4 m + 2 , then G is a ( P ≥ 3 , m ) ‐factor deleted graph. It's proved that this conjecture holds. However, it is found that as the connectivity changes, the tight lower bound of isolated toughness for ( P ≥ 3 , m ) ‐factor deleted graphs will change as well. Therefore, we propose a new perspective to look into this problem and introduce the concepts of isolated toughness ( P ≥ 3 , m ) factor deleted surface and isolated toughness variant ( P ≥ 3 , m ) factor deleted surface, where the result of the original conjecture is only a cross‐section on surfaces. The main contribution in this paper is to determine the concrete expression of these two surfaces.

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“…Some recent advances in graph factors can be referred to Zhou et al [6] and Gao et al [7][8][9]. e main contribution in this work is to study the existence of P ≥2 -factor or P ≥3 -factor in special settings by means of sun toughness parameter.…”

Section: Introductionmentioning

confidence: 99%

Sun Toughness Conditions for $P_{2}$ and

Gong

Baskonus

Gao

2021

Journal of Mathematics

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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.

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Toughness condition for a graph to be all fractional (g,f,n)-critical deleted

Cited by 24 publications

References 32 publications

Tight bounds for the existence of path factors in network vulnerability parameter settings

Tight bounds for the existence of path factors in network vulnerability parameter settings

Network vulnerability parameter and results on two surfaces

Sun Toughness Conditions for $P_{2}$ and

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Toughness condition for a graph to be all fractional (g,f,n)-critical deleted

Cited by 24 publications

References 32 publications

Tight bounds for the existence of path factors in network vulnerability parameter settings

Tight bounds for the existence of path factors in network vulnerability parameter settings

Network vulnerability parameter and results on two surfaces

Sun Toughness Conditions for P 2 and

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Sun Toughness Conditions for $P_{2}$ and