scholarly journals On Computing the Distinguishing Numbers of Trees and Forests

10.37236/1037 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Christine T. Cheng
Keyword(s):  
Automorphism Groups ◽  
Combinatorial Properties ◽  
Distinguishing Number ◽  
Minimum Number

Let $G$ be a graph. A vertex labeling of $G$ is distinguishing if the only label-preserving automorphism of $G$ is the identity map. The distinguishing number of $G$, $D(G)$, is the minimum number of labels needed so that $G$ has a distinguishing labeling. In this paper, we present $O(n \log n)$-time algorithms that compute the distinguishing numbers of trees and forests. Unlike most of the previous work in this area, our algorithm relies on the combinatorial properties of trees rather than their automorphism groups to compute for their distinguishing numbers.

scholarly journals Bounding the Distinguishing Number of Infinite Graphs and Permutation Groups

10.37236/3046 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
Simon M. Smith ◽  
Mark E. Watkins
Keyword(s):  
Connected Graph ◽  
Automorphism Groups ◽  
Cardinal Number ◽  
Infinite Graph ◽  
Infinite Graphs ◽  
Finite Radius ◽  
Locally Finite ◽  
Distinguishing Number ◽  
Identity Permutation ◽  
Imprimitive Group

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.

Coinitial Grapfis and Whitehead Automorphisms

1979 ◽  
Vol 31 (1) ◽  
pp. 112-123 ◽  
Author(s):  
A. H. M. Hoare
Keyword(s):  
Automorphism Groups ◽  
Fuchsian Groups ◽  
Topological Methods ◽  
Combinatorial Properties ◽  
Algebraic Proofs

Coinitial graphs were used in [2; 3 ; 4] as a combinatorial tool in the Reidemeister- Schreier process in order to prove subgroup theorems for Fuchsian groups. Whitehead had previously introduced such graphs but used topological methods for his proofs [8; 9]. Subsequently Rapaport [7] and Iliggins and Lyndon [1] gave algebraic proofs of the results in [9], and AIcCool [5; 6] has further developed these methods so that presentations of automorphism groups could be found.In this paper it is shown that Whitehead automorphisms can be described by a “cutting and pasting” operation on coinitial graphs. Section 1 defines and gives some combinatorial properties of these operations, based on [1].

On Computing Component (Edge) Connectivities of Balanced Hypercubes

2019 ◽  
Vol 63 (9) ◽  
pp. 1311-1320 ◽  
Author(s):  
Mei-Mei Gu ◽  
Jou-Ming Chang ◽  
Rong-Xia Hao
Keyword(s):  
Upper Bound ◽  
Edge Connectivity ◽  
Combinatorial Properties ◽  
Minimum Number ◽  
Two Parameters ◽  
Disconnected Graph

Abstract For an integer $\ell \geqslant 2$, the $\ell $-component connectivity (resp. $\ell $-component edge connectivity) of a graph $G$, denoted by $\kappa _{\ell }(G)$ (resp. $\lambda _{\ell }(G)$), is the minimum number of vertices (resp. edges) whose removal from $G$ results in a disconnected graph with at least $\ell $ components. The two parameters naturally generalize the classical connectivity and edge connectivity of graphs defined in term of the minimum vertex-cut and the minimum edge-cut, respectively. The two kinds of connectivities can help us to measure the robustness of the graph corresponding to a network. In this paper, by exploring algebraic and combinatorial properties of $n$-dimensional balanced hypercubes $BH_n$, we obtain the $\ell $-component (edge) connectivity $\kappa _{\ell }(BH_n)$ ($\lambda _{\ell }(BH_n)$). For $\ell $-component connectivity, we prove that $\kappa _2(BH_n)=\kappa _3(BH_n)=2n$ for $n\geq 2$, $\kappa _4(BH_n)=\kappa _5(BH_n)=4n-2$ for $n\geq 4$, $\kappa _6(BH_n)=\kappa _7(BH_n)=6n-6$ for $n\geq 5$. For $\ell $-component edge connectivity, we prove that $\lambda _3(BH_n)=4n-1$, $\lambda _4(BH_n)=6n-2$ for $n\geq 2$ and $\lambda _5(BH_n)=8n-4$ for $n\geq 3$. Moreover, we also prove $\lambda _\ell (BH_n)\leq 2n(\ell -1)-2\ell +6$ for $4\leq \ell \leq 2n+3$ and the upper bound of $\lambda _\ell (BH_n)$ we obtained is tight for $\ell =4,5$.

CROSS RATIO GRAPHS

2001 ◽  
Vol 64 (2) ◽  
pp. 257-272 ◽  
Author(s):  
A. GARDINER ◽  
CHERYL E. PRAEGER ◽  
SANMING ZHOU
Keyword(s):  
Automorphism Groups ◽  
Projective Line ◽  
Cross Ratio ◽  
Transitive Graph ◽  
Combinatorial Properties ◽  
The Family ◽  
The Cross ◽  
Transitive Graphs ◽  
Ordered Pairs

A family of arc-transitive graphs is studied. The vertices of these graphs are ordered pairs of distinct points from a finite projective line, and adjacency is defined in terms of the cross ratio. A uniform description of the graphs is given, their automorphism groups are determined, the problem of isomorphism between graphs in the family is solved, some combinatorial properties are explored, and the graphs are characterised as a certain class of arc-transitive graphs. Some of these graphs have arisen as examples in studies of arc-transitive graphs with complete quotients and arc-transitive graphs which ‘almost cover’ a 2-arc transitive graph.

scholarly journals Bounds on the Distinguishing Chromatic Number

10.37236/177 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Karen L. Collins ◽  
Mark Hovey ◽  
Ann N. Trenk
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Automorphism Group ◽  
Chromatic Number ◽  
Automorphism Groups ◽  
Upper Bounds ◽  
Underlying Graph ◽  
Minimum Number

Collins and Trenk define the distinguishing chromatic number $\chi_D(G)$ of a graph $G$ to be the minimum number of colors needed to properly color the vertices of $G$ so that the only automorphism of $G$ that preserves colors is the identity. They prove results about $\chi_D(G)$ based on the underlying graph $G$. In this paper we prove results that relate $\chi_D(G)$ to the automorphism group of $G$. We prove two upper bounds for $\chi_D(G)$ in terms of the chromatic number $\chi(G)$ and show that each result is tight: (1) if Aut$(G)$ is any finite group of order $p_1^{i_1} p_2^{i_2} \cdots p_k^{i_k}$ then $\chi_D(G) \le \chi(G) + i_1 + i_2 \cdots + i_k$, and (2) if Aut$(G)$ is a finite and abelian group written Aut$(G) = {\Bbb Z}_{p_{1}^{i_{1}}}\times \cdots \times {\Bbb Z}_{p_{k}^{i_{k}}}$ then we get the improved bound $\chi_D(G) \le \chi(G) + k$. In addition, we characterize automorphism groups of graphs with $\chi_D(G) = 2$ and discuss similar results for graphs with $\chi_D(G)=3$.

scholarly journals A Finite Word Poset

10.37236/1607 ◽  
2000 ◽  
Vol 8 (2) ◽  
Author(s):  
Péter L. Erdős ◽  
Péter Sziklai ◽  
David C. Torney
Keyword(s):  
Automorphism Groups ◽  
Partial Ordering ◽  
Finite Alphabet ◽  
Combinatorial Properties ◽  
Normalized Matching Property ◽  
Finite Word

Our word posets have finite words of bounded length as their elements, with the words composed from a finite alphabet. Their partial ordering follows from the inclusion of a word as a subsequence of another word. The elemental combinatorial properties of such posets are established. Their automorphism groups are determined (along with similar result for the word poset studied by Burosch, Frank and Röhl [4]) and a BLYM inequality is verified (via the normalized matching property).

scholarly journals On the Distance Pattern Distinguishing Number of a Graph

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Sona Jose ◽  
Germina K. Augustine
Keyword(s):  
Nonempty Subset ◽  
Simple Graph ◽  
Distinguishing Number ◽  
Minimum Number

LetG=(V,E)be a connected simple graph and letMbe a nonempty subset ofV. TheM-distance pattern of a vertexuinGis the set of all distances fromuto the vertices inM. If the distance patterns of all vertices inVare distinct, then the setMis a distance pattern distinguishing set ofG. A graphGwith a distance pattern distinguishing set is called a distance pattern distinguishing graph. Minimum number of vertices in a distance pattern distinguishing set is called distance pattern distinguishing number of a graph. This paper initiates a study on the problem of finding distance pattern distinguishing number of a graph and gives bounds for distance pattern distinguishing number. Further, this paper provides an algorithm to determine whether a graph is a distance pattern distinguishing graph or not and hence to determine the distance pattern distinguishing number of that graph.

On Component Connectivity of Hierarchical Star Networks

2020 ◽  
Vol 31 (03) ◽  
pp. 313-326
Author(s):  
Mei-Mei Gu ◽  
Jou-Ming Chang ◽  
Rong-Xia Hao
Keyword(s):  
Fault Tolerance ◽  
Network Topology ◽  
Interconnection Network ◽  
Building Blocks ◽  
Star Graphs ◽  
Combinatorial Properties ◽  
Star Networks ◽  
Minimum Number ◽  
Text Component ◽  
Disconnected Graph

For an integer [Formula: see text], the [Formula: see text]-component connectivity of a graph [Formula: see text], denoted by [Formula: see text], is the minimum number of vertices whose removal from [Formula: see text] results in a disconnected graph with at least [Formula: see text] components or a graph with fewer than [Formula: see text] vertices. This naturally generalizes the classical connectivity of graphs defined in term of the minimum vertex-cut. This kind of connectivity can help us to measure the robustness of the graph corresponding to a network. The hierarchical star networks [Formula: see text], proposed by Shi and Srimani, is a new level interconnection network topology, and uses the star graphs as building blocks. In this paper, by exploring the combinatorial properties and fault-tolerance of [Formula: see text], we study the [Formula: see text]-component connectivity of hierarchical star networks [Formula: see text]. We obtain the results: [Formula: see text], [Formula: see text] and [Formula: see text] for [Formula: see text].

scholarly journals Distinguishing Numbers for Graphs and Groups

10.37236/1816 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Julianna Tymoczko
Keyword(s):  
Group Action ◽  
General Problem ◽  
Group Actions ◽  
Graph Automorphism ◽  
Arbitrary Group ◽  
Distinguishing Number ◽  
Minimum Number ◽  
Trivial Graph ◽  
Open Question

A graph $G$ is distinguished if its vertices are labelled by a map $\phi: V(G) \longrightarrow \{1,2,\ldots, k\}$ so that no non-trivial graph automorphism preserves $\phi$. The distinguishing number of $G$ is the minimum number $k$ necessary for $\phi$ to distinguish the graph. It measures the symmetry of the graph. We extend these definitions to an arbitrary group action of $\Gamma$ on a set $X$. A labelling $\phi: X \longrightarrow \{1,2,\ldots,k\}$ is distinguishing if no element of $\Gamma$ preserves $\phi$ except those which fix each element of $X$. The distinguishing number of the group action on $X$ is the minimum $k$ needed for $\phi$ to distinguish the group action. We show that distinguishing group actions is a more general problem than distinguishing graphs. We completely characterize actions of $S_n$ on a set with distinguishing number $n$, answering an open question of Albertson and Collins.

scholarly journals Bounds for Distinguishing Invariants of Infinite Graphs

10.37236/6362 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Wilfried Imrich ◽  
Rafał Kalinowski ◽  
Monika Pilśniak ◽  
Mohammad Hadi Shekarriz
Keyword(s):  
Chromatic Number ◽  
Lower Bounds ◽  
Maximum Degree ◽  
Infinite Graphs ◽  
Chromatic Index ◽  
Distinguishing Number ◽  
Minimum Number ◽  
Edge Colourings ◽  
Delta G ◽  
Vertex Colouring

We consider infinite graphs. The distinguishing number $D(G)$ of a graph $G$ is the minimum number of colours in a vertex colouring of $G$ that is preserved only by the trivial automorphism. An analogous invariant for edge colourings is called the distinguishing index, denoted by $D'(G)$. We prove that $D'(G)\leq D(G)+1$. For proper colourings, we study relevant invariants called the distinguishing chromatic number $\chi_D(G)$, and the distinguishing chromatic index $\chi'_D(G)$, for vertex and edge colourings, respectively. We show that $\chi_D(G)\leq 2\Delta(G)-1$ for graphs with a finite maximum degree $\Delta(G)$, and we obtain substantially lower bounds for some classes of graphs with infinite motion. We also show that $\chi'_D(G)\leq \chi'(G)+1$, where $\chi'(G)$ is the chromatic index of $G$, and we prove a similar result $\chi''_D(G)\leq \chi''(G)+1$ for proper total colourings. A number of conjectures are formulated.

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