An Improvement to Chvátal and Thomassen’s Upper Bound for Oriented Diameter

Babu, Jasine; Benson, Deepu; Rajendraprasad, Deepak; Vaka, Sai Nishanth

doi:10.1007/978-3-030-50026-9_8

Cited by 3 publications

(3 citation statements)

References 9 publications

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“…Kwok, Liu and West [5] proved that 9 ≤ ḡ(3) ≤ 11. Together with Vaka, we showed that ḡ(4) ≤ 21 [6]. The major effort in the same paper for us was to show that ḡ(d) ≤ 1.373d 2 + 6.971d − 1.…”

Section: Further Literaturementioning

confidence: 79%

Improved Bounds for the Oriented Radius of Mixed Multigraphs

Babu¹,

Benson²,

Rajendraprasad³

2021

Preprint

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A mixed multigraph is a multigraph which may contain both undirected and directed edges. An orientation of a mixed multigraph G is an assignment of exactly one direction to each undirected edge of G. A mixed multigraph G can be oriented to a strongly connected digraph if and only if G is bridgeless and strongly connected [Boesch and Tindell, Am. Math. Mon., 1980]. For each r ∈ N, let f (r) denote the smallest number such that any strongly connected bridgeless mixed multigraph with radius r can be oriented to a digraph of radius at most f (r). We improve the current best upper bound of 4r 2 + 4r on f (r) [Chung, Garey and Tarjan, Networks, 1985] to 1.5r 2 + r + 1. Our upper bound is tight upto a multiplicative factor of 1.5 since, ∀r ∈ N, there exists an undirected bridgeless graph of radius r such that every orientation of it has radius at least r 2 + r [Chvátal and Thomassen, J. Comb. Theory. Ser. B., 1978]. We prove a marginally better lower bound, f (r) ≥ r 2 + 3r + 1, for mixed multigraphs. While this marginal improvement does not help with asymptotic estimates, it clears a natural suspicion that, like undirected graphs, f (r) may be equal to r 2 + r even for mixed multigraphs. En route, we show that if each edge of G lies in a cycle of length at most η, then the oriented radius of G is at most 1.5rη. All our proofs are constructive and lend themselves to polynomial time algorithms.

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Section: Further Literaturementioning

confidence: 79%

Improved Bounds for the Oriented Radius of Mixed Multigraphs

Babu¹,

Benson²,

Rajendraprasad³

2021

Preprint

Self Cite

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“…and constructed bridgeless graphs of diameter d for which every strong orientation admits a diameter of at least 1 2 d 2 + d. The upper bound was improved by Babu, Benson, Rajendraprasad and Vaka [1] to 1.373d 2 + 6.971d − 1.…”

Section: Definitionsmentioning

confidence: 99%

Large Girth and Small Oriented Diameter Graphs

Cochran¹

2022

Preprint

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In 2015, Dankelmann and Bau proved that for every bridgeless graph G of order n and minimum degree δ there is an orientation of diameter at most 11 n δ+1 + 9. In 2016, Surmacs reduced this bound to 7 n δ+1 . In this paper, we consider the girth of a graph g and show that for any ε > 0 there is a bound of the form (2g + ε) n h(δ,g) + O(1), where h(δ, g) is a polynomial. Letting g = 3 and ε < 1 gives an inprovement on the result by Surmacs.

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“…The upper bound in Theorem 1 is not sharp; an improvement of this bound was given by Babu et al [1]. Because it is very difficult to determine

\overset{d i a m}{stackreltrue} (G)

, researchers have tried to obtain sharp upper bounds for some special classes of graphs, such as complete

k

‐partite graphs, or to establish tight upper bounds in terms of other graph parameters, such as the domination number, minimum degree, maximum degree, and so on.…”

Section: Introductionmentioning

confidence: 99%

Oriented diameter of maximal outerplanar graphs

Wang

Chen

Dankelmann

et al. 2021

Journal of Graph Theory

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Let G be a finite connected undirected graph and stackreltrueG ⇀ a strong orientation of G. The diameter of stackreltrueG ⇀, denoted by d i a m ( G goodbreaktrue⇀ ), is the maximum directed distance between any two vertices of stackreltrueG ⇀. The oriented diameter of G is defined as stackreltrued i a m ⇀ ( G ) = m i n { d i a m ( G goodbreaktrue⇀ ) ∣ stackreltrueG ⇀nbsp0.25em is a strong orientation of G } . In this paper, we show that stackreltrued i a m ⇀ ( G ) ≤ ⌈ n ∕ 2 ⌉ for any maximal outerplanar graph G of order n ≥ 3, with four exceptions, and the upper bound is sharp.

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An Improvement to Chvátal and Thomassen’s Upper Bound for Oriented Diameter

Cited by 3 publications

References 9 publications

Improved Bounds for the Oriented Radius of Mixed Multigraphs

Improved Bounds for the Oriented Radius of Mixed Multigraphs

Large Girth and Small Oriented Diameter Graphs

Oriented diameter of maximal outerplanar graphs

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