Isolation of k-cliques II

Borg, Peter; Fenech, Kurt; Kaemawichanurat, Pawaton

doi:10.1016/j.disc.2021.112641

Cited by 17 publications

(8 citation statements)

References 6 publications

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“…Caro and Hansberg [8] proved that if G is a connected n-vertex graph with n ≥ 3, then ι(G, K 2 ) ≤ n/3 unless G is a 5-cycle. Solving a problem of Caro and Hansberg [8], Fenech, Kaemawichanurat and the present author [2] established the general sharp bound on ι(G, K k ) in Theorem 1.…”

mentioning

confidence: 69%

“…Theorem 4 generalizes Theorems 1-3 (as k-cliques are (k − 1)-regular and cycles are 2-regular), and the argument in its proof is more efficient than those in [1,2] (for Theorems 1 and 2). By Brooks' Theorem, Theorem 4 immediately yields the following generalization.…”

Section: Construction 1 ([2]mentioning

confidence: 82%

“…It is worth mentioning that the authors of [2] also obtained a sharp upper bound in terms of the number of edges [3]. Ore's result and the Caro-Hansberg result are the cases k = 1 and k = 2 of Theorem 1, respectively.…”

Section: Construction 1 ([2]mentioning

confidence: 88%

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Isolation of Cycles

Borg

2020

Graphs and Combinatorics

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mentioning

confidence: 69%

Section: Construction 1 ([2]mentioning

confidence: 82%

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Isolation of Cycles

Borg

2020

Graphs and Combinatorics

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“…Proposition 2.1. [9] If 𝑑 represents a diagonal of a mop 𝐺, it divides 𝐺 into two separate mops, namely 𝐺 1 and 𝐺 2 .…”

Section: Resultsmentioning

confidence: 99%

“…The theory of graph control has been an important and popular research direction in graph theory, receiving extensive attention and recognition from scholars. Currently, numerous research results have been achieved in the field of control theory [7][8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

Double-Star Isolation of Maximal Outerplanar Graphs ∗

Cao,

Zhang,

et al. 2023

J. Phys.: Conf. Ser.

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In the graph G = (V, E), V represents the set of vertices and E represents the set of edges. ℱ represents a family of graphs. A subset S ⊆ V is considered an ℱ -isolating set if G[V\NG[S]] does not contain F as a subgraph for all F ∈ ℱ. The ℱ-isolation number of the graph G, denoted by 𝜄(G, ℱ), is defined as the minimum cardinality of an ℱ-isolating set in G. If the family ℱ consists of a single graph H, then the subset S is referred to as an H-isolating set. The H-isolation number of the graph G, denoted by 𝜄 H (G), is defined as the minimum cardinality of an H-isolating set in G. Maximal outerplanar graphs have been widely applied in different fields of research since 1891, where they hold significant importance. This paper aims to provide a comprehensive analysis of the H-isolation number in maximal outerplanar graphs, ι H ( G ) ≤ min { n 2 k + 4 , n + n 2 2 k + 5 , n − n 2 2 k + 2 } , where H ≅ S 1,k+1,k+1.

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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 II

Cited by 17 publications

References 6 publications

Isolation of Cycles

Isolation of Cycles

Double-Star Isolation of Maximal Outerplanar Graphs ∗

Isolation of Regular Graphs and k-Chromatic Graphs

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