Several parameters of generalized Mycielskians

Lin, Wensong; Wu, Jianzhuan; Lam, Peter Che Bor; Gu, Guohua

doi:10.1016/j.dam.2005.11.001

Cited by 34 publications

(9 citation statements)

References 16 publications

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“…As seen in Theorem 1.1 and [18,5], the presence of isolated vertices in G, and therefore in µ (t) (G), has a significant effect on the structure and behavior of µ (t) (G). If v i is an isolated vertex in G and t ≥ 2, then in…”

Section: The Cost Of 2-distinguishing G Is Denoted ρ(G)mentioning

confidence: 90%

Determining Number and Cost of Generalized Mycielskian Graphs

Boutin¹,

Cockburn²,

Keough³

et al. 2020

Preprint

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A set S of vertices is a determining set for a graph G if every automorphism of G is uniquely determined by its action on S. The size of a smallest determining set for G is called its determining number, Det(G). A graph G is said to be d-distinguishable if there is a coloring of the vertices with d colors so that only the trivial automorphism preserves the color classes. The smallest d for which G is d-distinguishable is its distinguishing number, Dist(G). If Dist(G) = 2, the cost of 2-distinguishing, ρ(G), is the size of a smallest color class over all 2-distinguishing colorings of G. The Mycielskian, µ(G), of a graph G is constructed by adding a shadow master vertex w, and for each vertex vi of G adding a shadow vertex ui, with edges so that the neighborhood of ui in µ(G) is the same as the neighborhood of vi in G with the addition of w. That is, N (ui) = NG(vi) ∪ {w}. The generalized Mycielskian µ (t) (G) of a graph G is a Mycielskian graph with t layers of shadow vertices, each with edges to layers above and below, and the shadow master only adjacent to the top layer of shadow vertices. A graph is twin-free if it has no pair of vertices with the same set of neighbors. This paper examines the determining number and, when relevant, the cost of 2-distinguishing for Mycielskians and generalized Mycielskians of simple graphs with no isolated vertices. In particular, if G = K2 is twin-free with no isolated vertices, then Det(µ (t) For G with twins, we develop a framework using quotient graphs with respect to equivalence classes of twin vertices to give bounds on the determining number of Mycielskians. Moreover, we identify classes of graphs with twins for which Det(µ (t) (G)) = (t+1) Det(G).

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Section: The Cost Of 2-distinguishing G Is Denoted ρ(G)mentioning

confidence: 90%

Determining Number and Cost of Generalized Mycielskian Graphs

Boutin¹,

Cockburn²,

Keough³

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…The following consequence to Theorem 2.1 was first proven in [25]. More precisely, it can be deduced from Theorems 4.1 and 4.2 of [25] by specializing to the case m = 1 and by replacing the vertex cover number with the independence number.…”

Section: Independence and Packing Chromatic Number Of The Mycielskianmentioning

confidence: 91%

“…We present a formula for establishing α(M (G)) in an arbitrary graph G; to the best of our knowledge, this has not yet been established in full generality, cf. [25]. The obtained formula is then applied to obtain various bounds on χ ρ (M (G)).…”

Section: Introductionmentioning

confidence: 99%

Packing chromatic number versus chromatic and clique number

et al. 2017

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The packing chromatic number χ ρ (G) of a graph G is the smallest integer k such that the vertex set of G can be partitioned into sets V i , i ∈ [k], where each V i is an i-packing. In this paper, we investigate for a given triple (a, b, c) of positive integers whether there exists a graph G such that ω(G) = a, χ(G) = b, and χ ρ (G) = c. If so, we say that (a, b, c) is realizable. It is proved that b = c ≥ 3 implies a = b, and that triples (2, k, k + 1) and (2, k, k + 2) are not realizable as soon as k ≥ 4. Some of the obtained results are deduced from the bounds proved on the packing chromatic number of the Mycielskian. Moreover, a formula for the independence number of the Mycielskian is given. A lower bound on χ ρ (G) in terms of ∆(G) and α(G) is also proved.

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“…Concerning on the other results for (generalized) Mycielskian graph, one can refer to papers [15][16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

Reformulated Reciprocal Degree Distance and Reciprocal Degree Distance of the Complement of the Mycielskian Graph and Generalized Mycielskian

Fang

Bian

et al. 2019

Mathematical Problems in Engineering

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The reformulated reciprocal degree distance is defined for a connected graph G as R¯t(G)=(1/2)∑u,υ∈VG((dG(u)+dG(υ))/(dG(u,υ)+t)),t≥0, which can be viewed as a weight version of the t-Harary index; that is, H¯t(G)=(1/2)∑u,υ∈VG(1/(dG(u,υ)+t)),t≥0. In this paper, we present the reciprocal degree distance index of the complement of Mycielskian graph and generalize the corresponding results to the generalized Mycielskian graph.

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Several parameters of generalized Mycielskians

Cited by 34 publications

References 16 publications

Determining Number and Cost of Generalized Mycielskian Graphs

Determining Number and Cost of Generalized Mycielskian Graphs

Packing chromatic number versus chromatic and clique number

Reformulated Reciprocal Degree Distance and Reciprocal Degree Distance of the Complement of the Mycielskian Graph and Generalized Mycielskian

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