Base size, metric dimension and other invariants of groups and graphs

Bailey, Robert F.; Cameron, Peter J‎.

doi:10.1112/blms/bdq096

Cited by 225 publications

(308 citation statements)

References 78 publications

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“…It is easily verified that this inequality holds for all q ≥ 5 when n = 2, and also holds when (q, n) = (4, 3), (3,4), or (2,8). Also for fixed q, the left side of this inequality increases with n more rapidly than the right side, and so the result follows.…”

Section: Vector Spaces and Groupsmentioning

confidence: 98%

“…It has been shown by Cameron, Neumann and Saxl [7,6] that all but finitely many primitive permutation groups other than A n or S n (in their standard actions) have distinguishing number 2, and the exceptions have been classified by Seress [19]; see also [4]. For imprimitive actions, Chan [9] gives examples of wreath products of groups with large distinguishing numbers; see also [22].…”

Section: 7]mentioning

confidence: 99%

“…Then for given r, there are only finitely many wreath products H K where |K| ≤ r and H is primitive but H = A n or S n , with distinguishing number greater than 2. For further discussion of minimum base size and distinguishing number for wreath products, see [4].…”

Section: 7]mentioning

confidence: 99%

“…This number, denoted by m(A, X), or simply m(A) when the context is clear, is more commonly known as the minimal degree of the permutation group induced by A on X. The survey paper [4] provides a number of examples of other recent concepts in graph theory that have had previous lives, under different names, in the theory of permutation groups.…”

Section: Introductionmentioning

confidence: 99%

“…For all maps with more than 10 vertices, the action of the automorphism group on the vertices has distinguishing number two (see [20]). All primitive permutation groups of degree n > 32 other than S n or A n have distinguishing number two (see [19,4]). …”

Section: Introductionmentioning

confidence: 99%

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Motion and distinguishing number two

Conder

Tucker

2011

Ars Math. Contemp.

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A group A acting faithfully on a finite set X is said to have distinguishing number two if there is a proper subset Y whose (setwise) stabilizer is trivial. The motion of A acting on X is defined as the largest integer k such that all non-trivial elements of A move at least k elements of X. The Motion Lemma of Russell and Sundaram states that if the motion is at least 2 log 2 |A|, then the action has distinguishing number two. When X is a vector space, group, or map, the Motion Lemma and elementary estimates of the motion together show that in all but finitely many cases, the action of Aut(X) on X has distinguishing number two. A new lower bound for the motion of any transitive action gives similar results for transitive actions with restricted point-stabilizers. As an instance of what can happen with intransitive actions, it is shown that if X is a set of points on a closed surface of genus g, and |X| is sufficiently large with respect to g, then any action on X by a finite group of surface homeomorphisms has distinguishing number two.

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Section: Vector Spaces and Groupsmentioning

confidence: 98%

Section: 7]mentioning

confidence: 99%

Section: 7]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Motion and distinguishing number two

Conder

Tucker

2011

Ars Math. Contemp.

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Centroidal bases in graphs

2014

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We introduce the notion of a centroidal locating set of a graph G, that is, a set L of vertices such that all vertices in G are uniquely determined by their relative distances to the vertices of L. A centroidal locating set of G of minimum size is called a centroidal basis, and its size is the centroidal dimension CD(G). This notion, which is related to previous concepts, gives a new way of identifying the vertices of a graph. The centroidal dimension of a graph G is lowerand upper-bounded by the metric dimension and twice the location-domination number of G, respectively. The latter two parameters are standard and well-studied notions in the field of graph identification.We show that for any graph G with n vertices and maximum degree at least 2, (1 + o(1)) ln n ln ln n ≤ CD(G) ≤ n − 1. We discuss the tightness of these bounds and in particular, we characterize the set of graphs reaching the upper bound. We then show that for graphs in which every pair of vertices is connected via a bounded number of paths, CD(G) = Ω |E(G)| , the bound being tight for paths and cycles. We finally investigate the computational complexity of determining CD(G) for an input graph G, showing that the problem is hard and cannot even be approximated efficiently up to a factor of o(log n). We also give an O √ n ln napproximation algorithm.

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A simple algorithm for graph reconstruction

Mathieu

Zhou

2023

Random Struct Algorithms

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How efficiently can we find an unknown graph using distance queries between its vertices? We assume that the unknown graph is connected, unweighted, and has bounded degree. The goal is to find every edge in the graph. This problem admits a reconstruction algorithm based on multi‐phase Voronoi‐cell decomposition and using Õfalse(n3false/2false)$$ \overset{\widetilde }{O}\left({n}^{3/2}\right) $$ distance queries. In our work, we analyze a simple reconstruction algorithm. We show that, on random normalΔ$$ \Delta $$‐regular graphs, our algorithm uses Õfalse(nfalse)$$ \overset{\widetilde }{O}(n) $$ distance queries. As by‐products, with high probability, we can reconstruct those graphs using log2n$$ {\log}^2n $$ queries to an all‐distances oracle or Õfalse(nfalse)$$ \overset{\widetilde }{O}(n) $$ queries to a betweenness oracle, and we bound the metric dimension of those graphs by log2n$$ {\log}^2n $$. Our reconstruction algorithm has a very simple structure, and is highly parallelizable. On general graphs of bounded degree, our reconstruction algorithm has subquadratic query complexity.

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Base size, metric dimension and other invariants of groups and graphs

Cited by 225 publications

References 78 publications

Motion and distinguishing number two

Motion and distinguishing number two

Centroidal bases in graphs

A simple algorithm for graph reconstruction

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