Trees with distinguishing index  equal  distinguishing number plus one

A group of permutations $G$ of a set $V$ is $k$-distinguishable if there exists a partition of $V$ into $k$ cells such that only the identity permutation in $G$ fixes setwise all of the cells of the partition. The least cardinal number $k$ such that $(G,V)$ is $k$-distinguishable is its distinguishing number $D(G,V)$. In particular, a graph $\Gamma$ is $k$-distinguishable if its automorphism group $\rm{Aut}(\Gamma)$ satisfies $D(\rm{Aut}(\Gamma),V\Gamma)\leq k$.Various results in the literature demonstrate that when an infinite graph fails to have some property, then often some finite subgraph is similarly deficient. In this paper we show that whenever an infinite connected graph $\Gamma$ is not $k$-distinguishable (for a given cardinal $k$), then it contains a ball of finite radius whose distinguishing number is at least $k$. Moreover, this lower bound cannot be sharpened, since for any integer $k \geq 3$ there exists an infinite, locally finite, connected graph $\Gamma$ that is not $k$-distinguishable but in which every ball of finite radius is $k$-distinguishable.In the second half of this paper we show that a large distinguishing number for an imprimitive group $G$ is traceable to a high distinguishing number either of a block of imprimitivity or of the induced action by $G$ on the corresponding system of imprimitivity. An immediate application is to automorphism groups of infinite imprimitive graphs. These results are companion to the study of the distinguishing number of infinite primitive groups and graphs in a previous paper by the authors together with T. W. Tucker.

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Distinguishability of Locally Finite Trees

The Electronic Journal of Combinatorics ◽

10.37236/947 ◽

2007 ◽

Vol 14 (1) ◽

Cited By ~ 11

Author(s):

Mark E. Watkins ◽

Xiangqian Zhou

Keyword(s):

Positive Integer ◽

Chromatic Number ◽

Inverse Image ◽

Vertex Coloring ◽

Finite Tree ◽

Locally Finite ◽

Distinguishing Number ◽

Nonidentity Element ◽

Locally Countable ◽

Proper Vertex

The distinguishing number $\Delta(X)$ of a graph $X$ is the least positive integer $n$ for which there exists a function $f:V(X)\to\{0,1,2,\cdots,n-1\}$ such that no nonidentity element of $\hbox{Aut}(X)$ fixes (setwise) every inverse image $f^{-1}(k)$, $k\in\{0,1,2,\cdots,n-1\}$. All infinite, locally finite trees without pendant vertices are shown to be 2-distinguishable. A proof is indicated that extends 2-distinguishability to locally countable trees without pendant vertices. It is shown that every infinite, locally finite tree $T$ with finite distinguishing number contains a finite subtree $J$ such that $\Delta(J)=\Delta(T)$. Analogous results are obtained for the distinguishing chromatic number, namely the least positive integer $n$ such that the function $f$ is also a proper vertex-coloring.

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Symmetry Breaking in Graphs

The Electronic Journal of Combinatorics ◽

10.37236/1242 ◽

1996 ◽

Vol 3 (1) ◽

Cited By ~ 76

Author(s):

Michael O. Albertson ◽

Karen L. Collins

Keyword(s):

Symmetry Breaking ◽

Complete Graph ◽

Subject Classification ◽

Distinguishing Number

A labeling of the vertices of a graph G, $\phi :V(G) \rightarrow \{1,\ldots,r\}$, is said to be $r$-distinguishing provided no automorphism of the graph preserves all of the vertex labels. The distinguishing number of a graph G, denoted by $D(G)$, is the minimum $r$ such that $G$ has an $r$-distinguishing labeling. The distinguishing number of the complete graph on $t$ vertices is $t$. In contrast, we prove (i) given any group $\Gamma$, there is a graph $G$ such that $Aut(G) \cong \Gamma$ and $D(G)= 2$; (ii) $D(G) = O(log(|Aut(G)|))$; (iii) if $Aut(G)$ is abelian, then $D(G) \leq 2$; (iv) if $Aut(G)$ is dihedral, then $D(G) \leq 3$; and (v) If $Aut(G) \cong S_4$, then either $D(G) = 2$ or $D(G) = 4$. Mathematics Subject Classification 05C,20B,20F,68R

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The distinguishing number and distinguishing index of the lexicographic product of two graphs

Discussiones Mathematicae Graph Theory ◽

10.7151/dmgt.2070 ◽

2018 ◽

Vol 38 (3) ◽

pp. 853 ◽

Cited By ~ 2

Author(s):

Saeid Alikhani ◽

Samaneh Soltani

Keyword(s):

Lexicographic Product ◽

Distinguishing Number

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DISTINGUISHING NUMBER AND DISTINGUISHING INDEX OF JOIN OF TWO SPECIFIC GRAPHS

Advances and Applications in Discrete Mathematics ◽

10.17654/dm017040467 ◽

2016 ◽

Vol 17 (4) ◽

pp. 467-485

Author(s):

Saeid Alikhani ◽

Samaneh Soltani ◽

Abdul Jalil M. Khalaf

Keyword(s):

Distinguishing Number

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Distinguishing number and distinguishing index of certain graphs

Filomat ◽

10.2298/fil1714393a ◽

2017 ◽

Vol 31 (14) ◽

pp. 4393-4404 ◽

Cited By ~ 3

Author(s):

Saeid Alikhani ◽

Samaneh Soltani

Keyword(s):

Distinguishing Number ◽

Two Parameters ◽

Corona Product ◽

Edge Labeling

The distinguishing number (index) D(G) (D0(G)) of a graph G is the least integer d such that G has an vertex labeling (edge labeling) with d labels that is preserved only by a trivial automorphism. In this paper we compute these two parameters for some specific graphs. Also we study the distinguishing number and the distinguishing index of corona product of two graphs.

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