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# Structure and Properties of Eccentric Digraphs - PowerPoint PPT Presentation

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

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

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

• Converse

• Symmetry (eccentric graph)

• Complement

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

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 ComplementsThe symmetric case

ED(G) = G when

G is self centered

G is disconnected

with each component

a complete graph

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

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

G

G–

• 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

Digraph

Iteration

Sequence

Digraph

Iteration

Sequence

Digraph

Iteration

Sequence

G

t=3

ED(G)

ED2(G)

ED3(G)

p=2

ED4(G)

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

• How long can the tail be?

• What can be the period?

• Iso-tail?

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

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

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).

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

A digraph G is eccentric if and only if

ED(G–) = G

G

G–

ED(G–)

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

Recall

p(Km Kn) = p(Km,n) = 2,

t(Km Kn) = t(Km,n) = 0

p(Hm Hn) = 2,

t(Hm Hn) = 1

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)

3

2

2

3

3

3

2

3

2

3

3

3

2

3

3

2

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

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

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

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

The graph C9 and its iterated

eccentric (di)graphs

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

• 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?