Isolation of Cycles

Borg, Peter

doi:10.1007/s00373-020-02143-2

Cited by 34 publications

(7 citation statements)

References 14 publications

(26 reference statements)

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“…Let C denote the set of cycles. The author [1] obtained a sharp upper bound on ι(G, C), and consequently settled another problem of Caro and Hansberg [3]. Before stating the result, we recall the explicit construction used for an extremal case.…”

Section: Introductionmentioning

confidence: 92%

“…We prove Theorems 3 and 4 using the same new strategy. We build on key ideas from [1], but, out of necessity, we provide a more efficient inductive argument and new ingredients, which are more abundant in the proof of Theorem 3.…”

Section: Moreover Equality Holds Ifmentioning

confidence: 99%

“…We start with two lemmas from [1] that will be used repeatedly. We provide their proofs for completeness.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Now suppose H 1 , H 2 ∈ H (1) ∪ H (3) . For any H ≃ P 3 , let y 1 H and y 3 H be the two leaves of H, and let y 2 H be the remaining vertex of H (so (3) , then by applying the argument for the previous case, we obtain that…”

Section: Subsubcase 222: For Eachmentioning

confidence: 99%

“…For standard terminology in graph theory, we refer the reader to [11]. Most of the notation and terminology used here is defined in [1], which lays the foundation for the work presented here.…”

Section: Introductionmentioning

confidence: 99%

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Isolation of connected graphs

Borg¹

2021

Preprint

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For a connected n-vertex graph G and a set F of graphs, let ι(G, F) denote the size of a smallest set D of vertices of G such that the graph obtained from G by deleting the closed neighbourhood of D contains no graph in F. Let E k denote the set of connected graphs that have at least k edges. By a result of Caro and Hansberg, ι(G, E 1 ) ≤ n/3 if n = 2 and G is not a 5-cycle. The author recently showed that if G is not a triangle and C is the set of cycles, then ι(G, C) ≤ n/4. We improve this result by showing that ι(G, E 3 ) ≤ n/4 if G is neither a triangle nor a 7-cycle. Let r be the number of vertices of G that have only one neighbour. We determine a set S of six graphs such that ι(G, E 2 ) ≤ (4n − r)/14 if G is not a copy of a member of S. The bounds are sharp.

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Section: Introductionmentioning

confidence: 92%

Section: Moreover Equality Holds Ifmentioning

confidence: 99%

“…We start with two lemmas from [1] that will be used repeatedly. We provide their proofs for completeness.…”

Section: Proof Of Theoremmentioning

confidence: 99%

Section: Subsubcase 222: For Eachmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Isolation of connected graphs

Borg¹

2021

Preprint

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Isolation of Regular Graphs and k-Chromatic Graphs

Borg

2024

Mediterr. J. Math.

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Given a set $${\mathcal {F}}$$ F of graphs, we call a copy of a graph in $${\mathcal {F}}$$ F an $${\mathcal {F}}$$ F -graph. The $${\mathcal {F}}$$ F -isolation number of a graph G, denoted by $$\iota (G,{\mathcal {F}})$$ ι ( G , F ) , is the size of a smallest set D of vertices of G such that the closed neighborhood of D intersects the vertex sets of the $${\mathcal {F}}$$ F -graphs contained by G (equivalently, $$G - N[D]$$ G - N [ D ] contains no $${\mathcal {F}}$$ F -graph). Thus, $$\iota (G,\{K_1\})$$ ι ( G , { K 1 } ) is the domination number of G. For any integer $$k \ge 1$$ k ≥ 1 , let $${\mathcal {F}}_{1,k}$$ F 1 , k be the set of regular graphs of degree at least $$k-1$$ k - 1 , let $${\mathcal {F}}_{2,k}$$ F 2 , k be the set of graphs whose chromatic number is at least k, and let $${\mathcal {F}}_{3,k}$$ F 3 , k be the union of $${\mathcal {F}}_{1,k}$$ F 1 , k and $${\mathcal {F}}_{2,k}$$ F 2 , k . Thus, k-cliques are members of both $${\mathcal {F}}_{1,k}$$ F 1 , k and $${\mathcal {F}}_{2,k}$$ F 2 , k . We prove that for each $$i \in \{1, 2, 3\}$$ i ∈ { 1 , 2 , 3 } , $$\frac{m+1}{{k \atopwithdelims ()2} + 2}$$ m + 1 k 2 + 2 is a best possible upper bound on $$\iota (G, {\mathcal {F}}_{i,k})$$ ι ( G , F i , k ) for connected m-edge graphs G that are not k-cliques. The bound is attained by infinitely many (non-isomorphic) graphs. The proof of the bound depends on determining the graphs attaining the bound. This appears to be a new feature in the literature on isolation. Among the result’s consequences are a sharp bound of Fenech, Kaemawichanurat, and the present author on the k-clique isolation number and a sharp bound on the cycle isolation number.

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