Structure and properties of eccentric digraphs
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Structure and Properties of Eccentric Digraphs. Joint work of Joan Gimbert Universitat de Lleida, Spain Nacho Lopez Universitat de Lleida, Spain Mirka Miller University of Ballarat, Australia Frank Ruskey University of Victoria, Canada

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Structure and properties of eccentric digraphs l.jpg
Structure and Properties of Eccentric Digraphs

Joint work of

  • Joan Gimbert Universitat de Lleida, Spain

  • Nacho Lopez Universitat de Lleida, Spain

  • Mirka Miller University of Ballarat, Australia

  • Frank Ruskey University of Victoria, Canada

  • Joe Ryan University of Ballarat, Australia


Eccentric digraph of a graph l.jpg
Eccentric Digraph of a Graph

eG(u) – the eccentricity of a vertex u in a graph G

v is an eccentric vertex of u if d(u,v) = e(u)

The eccentric digraph of G, ED(G)is a graph on the same vertex set as G but with an arc from u to v if and only if v is an eccentric vertex of u.

Buckley 2001



Eccentric digraphs and other graph and digraph operators l.jpg
Eccentric Digraphs and Other Graph and Digraph Operators

  • Converse

  • Symmetry (eccentric graph)

  • Complement


Eccentric digraphs and converse l.jpg
Eccentric Digraphs and Converse

For converse, just

change direction

of the arrows.

Let G be a digraph such that ED(G) = G, then

  • rad(G) > 1 unless G is a complete digraph,

  • G cannot have a digon unless G is a complete digraph,

  • ED2(G) = G


Symmetric eccentric digraphs l.jpg
Symmetric Eccentric Digraphs

For G a connected graph

ED(G) is symmetric  G is self centered

(Not true for digraphs

See C4, K3 K2 for examples)

For G not strongly connected digraph, ED(G) is symmetric 

G=H1H2 … Hk or

G=Kn→(H1 H2 …Hk)

Where H1, H2…are strongly connected components


Eccentric digraphs and complements the symmetric case l.jpg
Eccentric Digraphs and ComplementsThe symmetric case

ED(G) = G when

G is self centered

of radius 2

G is disconnected

with each component

a complete graph


Eccentric digraphs and complements the symmetric case8 l.jpg
Eccentric Digraphs and ComplementsThe symmetric case

The Even Cycle

ED(C6) = 3K2

ED2(C6) = H2,3

C6

C2n

ED(C2n) = nK2

ED2(C2n) = H2,n


Eccentric digraphs and complements l.jpg
Eccentric Digraphs and Complements

  • Construct G– (the reduction of G) by removing all out-arcs of v where out-deg(v) = n-1

G

G–


Eccentric digraphs and complements10 l.jpg
Eccentric Digraphs and Complements

  • Construct G– (the reduction of G) by removing all outarcs of v where deg(v) = n-1

  • Find G–, the complement of the reduction.

G

G–

G–

For a digraph G, ED(G) = G– if and only if,

for u V(G) with e(u) > 2, then

(u,v), (v,w)  E(G)  (u,w)  E(G)

v,w  V(G) and u≠ w


Slide11 l.jpg

An Eccentric

Digraph

Iteration

Sequence


Slide12 l.jpg

An Eccentric

Digraph

Iteration

Sequence


Slide13 l.jpg

An Eccentric

Digraph

Iteration

Sequence

G

t=3

ED(G)

ED2(G)

ED3(G)

p=2

ED4(G)


Isomorphisms l.jpg
Isomorphisms

For every digraph G there exist smallest integer numbers p' > 0 and t' 0 such that

EDt'(G)  EDp'+t'(G)

where  denotes graph isomorphism.

Call p' = p'(G) the iso-period and

t' = t'(G) the iso-tail.

Period = 2

Iso-period = 1


Questions l.jpg
Questions

  • How long can the tail be?

  • What can be the period?

  • What about the iso-period?

  • Iso-tail?

Theorem (Gimbert, Lopez, Miller, R; to appear)

For every digraph G, t(G) = t'(G)


Slide16 l.jpg

Finite – so there are digraphs that are not eccentric

digraphs for any other (di)graph.

How long can the tail be?

Digraphs containing a vertex with zero out degree

are not EDs

Theorem: (Boland, Buckley, Miller; 2004) Can construct an ED from a (di)graph by adding no more than one vertex (with appropriate arcs).


Characterisation of eccentric digraphs l.jpg
Characterisation of Eccentric Digraphs

Theorem (Gimbert, Lopez, Miller, R; to appear)

A digraph G is eccentric if and only if

ED(G–) = G

G

G–

ED(G–)


What can be the period l.jpg
What can be the period?

Computer searches over digraphs of up to 40 nodes indicate that for the most part

p(G) = 2

Theorem: (Wormald) Almost all digraphs have

iteration sequence period = 2


Period and iso period l.jpg
Period and Iso-period

Recall

p(Km Kn) = p(Km,n) = 2,

t(Km Kn) = t(Km,n) = 0


Period and iso period20 l.jpg
Period and Iso-period

p(Hm Hn) = 2,

t(Hm Hn) = 1



Period and tail of some families of graphs l.jpg
Period and Tail of Some Families of Graphs

  • Define Eccentric Core of G, EccCore(G) as the subdigraph of ED(G) induced by the vertices that in G are eccentric to some other vertex.

G

ED(G)

EccCore(G)


Slide23 l.jpg

3

3

2

2

3

3

3

2

3

2

3

3

3

2

3

3

2

3


Slide24 l.jpg

3

K2  K4

K2  K4

3

2

2

3

3

3

2

3

2

3

3

K2 C4

K2 C4

3

2

3

3

2

3


Slide25 l.jpg

R = The Cayley graph with generators

(01)(23)(4567) and (56)(78)


Slide26 l.jpg

A digraph G of order 10 such that

p(G) = p'(G) = 4

and t(G) = t'(G) = 1


Slide27 l.jpg

The graph C9 and its iterated

eccentric (di)graphs


Eccentric di graph period for odd cycles l.jpg
Eccentric (di)graph period for odd cycles

Sequence A003558 in Sloane’s Encyclopedia of Integer Sequences

p(C2m+1) = min{k>1: m(m+1)k-1 =  1 mod(2m+1)}

In particular, m = 2k, p(C2m+1) = k+1

Sloane’s A045639, the Queneau Numbers



Open problems l.jpg
Open Problems

  • Find the period and tail of various classes of graphs and digraphs.

  • What can be said about the size of the equivalence class in the labelled and unlabelled cases?