Isolation of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e25" altimg="si20.svg"><mml:mi>k</mml:mi></mml:math>-cliques

Borg, Peter; Fenech, Kurt; Kaemawichanurat, Pawaton

doi:10.1016/j.disc.2020.111879

Cited by 34 publications

(4 citation statements)

References 11 publications

(14 reference statements)

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“…They proved that if G is a connected n-vertex graph, then ι(G, K 2 ) ≤ n/3 unless G ≃ K 2 or G ≃ C 5 . Fenech, Kaemawichanurat, and the present author [2] generalized these bounds by showing that for any k…”

Section: Introductionmentioning

confidence: 70%

“…) by the induction hypothesis and Lemma 3(b). Note that, since H 1 ∈ H (2) ∪ H (4) ∪ H (5) ∪ H (6) , if y ∈ Y 1 such that y is a vertex of G − N[D 1 ], then H 1 ∈ H (5) ∪H (6) , y is the leaf of H 1 , and…”

Section: Subsubcase 222: For Eachmentioning

confidence: 99%

“…By the induction hypothesis, each member of H ′ is a copy of K 3 or of C 7 . Let H (1) = {H ∈ H ′ : H ≃ K 3 } and H (2) = {H ∈ H ′ : H ≃ C 7 }. Thus, H ′ = H (1) ∪ H (2) .…”

Section: Subsubcase 222: For Eachmentioning

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: 70%

Section: Subsubcase 222: For Eachmentioning

confidence: 99%

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Isolation of connected graphs

Borg¹

2021

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“…Borg et al [4] obtained that ι k (G) ≤ min{ n k+4 , n+n 2 5 , n−n 2 3 } for a maximal outerplanar graph G and k ≥ 0, where n 2 is the number of vertices of degree 2. Borg et al [7] studied that if [16] analyzed that ι(T ) = n 3 implies ι(T ) = γ(T ) for a tree T , and proposed simple algorithms to build these trees from the connections of stars.…”

Section: Introductionmentioning

confidence: 99%

Partial Domination of Network Models *

Zhang

Jia

Zhao

2022

Preprint

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Partial domination [8] is a theory first proposed in 2015 on the basis of domination theory, which has a wide range of research value. Let G = (V, E) be a graph and F be a family of graphs, a subset S ⊆ V is called an F-isolating set of G if G[V \NG[S]] does not contain F as a subgraph for all F ∈ F. If F = {K2}, S is an isolating set of G if G[V \NG[S]] does not contain K2. The isolation number of G is the minimum cardinality of an isolating set of G, denoted by ι(G). The hypercube network and n-star network are the basic models for interconnection networks, and they have many attractive topological properties. In this paper, we investigate the sharp bounds of the isolation numbers of the hypercube network Qn and n-star network Sn, and obtain (2n−1)\n ≤ ι(Qn) ≤ 2n−3 for any positive integer n ≥ 4 and (n·(n2−2)!)\2 ≤ ι(Sn) ≤ (n − 1)! for any positive integer n ≥ 2. AMS subject classification: 05C05, 05C12, 05C76

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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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Isolation of k-cliques

Cited by 34 publications

References 11 publications

Isolation of connected graphs

Isolation of connected graphs

Partial Domination of Network Models *

Isolation of Regular Graphs and k-Chromatic Graphs

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