Upper bounds on the Steiner diameter of a graph

Ali, Patrick; Dankelmann, Peter; Mukwembi, Simon

doi:10.1016/j.dam.2012.03.031

Cited by 31 publications

(17 citation statements)

References 3 publications

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“…Furthermore, Dankelmann et al [8,9] obtained additional bounds on the Steiner k-diameter related to the minimum degree and other graph invariants. Moreover, they constructed graphs to show that these bounds are asymptotically the best possible.…”

Section: Theorem 1 ([7])mentioning

confidence: 99%

A Note on the Steinerk-Diameter of Tensor Product Networks

Arunandhi¹,

Cheng

Melekian

2019

Parallel Process. Lett.

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Given a graph [Formula: see text] and [Formula: see text], the Steiner distance [Formula: see text] is the minimum size among all connected subgraphs of [Formula: see text] whose vertex sets contain [Formula: see text]. The Steiner [Formula: see text]-diameter [Formula: see text] is the maximum value of [Formula: see text] among all sets of [Formula: see text] vertices. In this short note, we study the Steiner [Formula: see text]-diameters of the tensor product of complete graphs.

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Section: Theorem 1 ([7])mentioning

confidence: 99%

A Note on the Steinerk-Diameter of Tensor Product Networks

Arunandhi¹,

Cheng

Melekian

2019

Parallel Process. Lett.

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“…Hence this definition yields a natural extension of the notion of “distance” for sets of more than two vertices. Corresponding notions of radius, diameter, and convexity have been studied in the literature [1, 2, 3, 4, 5, 10], see also the recent survey by Mao [12]. Day, Oellermann and Swart [7, 6] studied the Steiner distance within induced subgraphs, extending Howorka's concept of distance hereditary graphs .…”

Section: Introductionmentioning

confidence: 99%

Isometric subgraphs for Steiner distance

Weißauer

2020

Journal of Graph Theory

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Let G be a connected graph and  ℓ → E G : ( ) + a length-function on the edges of G. The Steiner distance sd G (A) of A ⊆ V(G) within G is the minimum length of a connected subgraph of G containing A,where the length of a subgraph is the sum of the lengths of its edges. It is clear that every subgraph. It is easy to construct examples of graphs H ⊆ G such that H is k-isometric, but not (k + 1)-isometric, so this defines a strict hierarchy of properties. We are interested in situations in which this hierarchy collapses in the sense that if H ⊆ G is k-isometric, then H is already fully isometric in G. Our first result of this kind asserts that if T is a tree and T ⊆ G is 2-isometric with respect to some lengthfunction ℓ, then it is fully isometric. This fails for graphs containing a cycle. We then prove that if C is a cycle and C ⊆ G is 6-isometric, then C is fully isometric. We present an example showing that the number 6 is indeed optimal. We then develop a structural approach toward a more general theory and present several open questions concerning the big picture underlying this phenomenon. K E Y W O R D Sgraph theory, isometry, steiner distance, steiner tree How to cite this article: Weißauer D. Isometric subgraphs for Steiner distance. J Graph Theory. 2020;94:597-613. https://doi.

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“…For S = {u, v}, d(S) = d(u, v) reduces to the classical distance between the two vertices u and v. Another basic observation is that if |S| = k, d(S) ≥ k − 1. For more results regarding varied properties of the Steiner distance, we refer to the reader to [3][4][5][6][7][8].…”

Section: Introductionmentioning

confidence: 99%

“…Given a graph G(V, E) and a vertex set S ⊆ V (G) containing no less than two vertices, an S-Steiner tree (or an S-tree, a Steiner tree connecting S) is defined as a subgraph T (V ′ , E ′ ) of G which is a subtree satisfying S ⊆ V ′ . If G is connected with order no less than 2 and S ⊆ V is nonempty, the Steiner distance d(S) among the vertices of S (sometimes simply put as the distance of S) is the minimum size of connected subgraph whose vertex sets contain the set S. For more results regarding varied properties of the Steiner distance, we refer to the reader to [1,5,6,10,12,24].…”

Section: Introductionmentioning

confidence: 99%

Nordhaus–Gaddum-Type Results for the Steiner Gutman Index of Graphs

et al. 2020

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Building upon the notion of the Gutman index SGut(G), Mao and Das recently introduced the Steiner Gutman index by incorporating Steiner distance for a connected graph G. The Steiner Gutman k-index SGutk(G) of G is defined by SGutk(G)=∑S⊆V(G),|S|=k∏v∈SdegG(v)dG(S), in which dG(S) is the Steiner distance of S and degG(v) is the degree of v in G. In this paper, we derive new sharp upper and lower bounds on SGutk, and then investigate the Nordhaus-Gaddum-type results for the parameter SGutk. We obtain sharp upper and lower bounds of SGutk(G)+SGutk(G¯) and SGutk(G)·SGutk(G¯) for a connected graph G of order n, m edges, maximum degree Δ and minimum degree δ.

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Upper bounds on the Steiner diameter of a graph

Cited by 31 publications

References 3 publications

A Note on the Steinerk-Diameter of Tensor Product Networks

A Note on the Steinerk-Diameter of Tensor Product Networks

Isometric subgraphs for Steiner distance

Nordhaus–Gaddum-Type Results for the Steiner Gutman Index of Graphs

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