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 GrapheG(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

),(max

)( vudGv

ue

A Graph and its Eccentric Digraph

Eccentric Digraphs and Other Graph and Digraph Operators

• Converse

• Symmetry (eccentric graph)

• Complement

Eccentric Digraphs and Converse

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

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

ii) G cannot have a digon unless G is a complete digraph,

iii) ED2(G) = G

For converse, justchange direction of the arrows.

Symmetric Eccentric DigraphsFor 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=H1H2 … 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 of radius 2

G is disconnectedwith each componenta complete graph

Eccentric Digraphs and ComplementsThe symmetric case

C6ED(C6) = 3K2 ED2(C6) = H2,3

C2n ED(C2n) = nK2 ED2(C2n) = H2,n

The Even Cycle

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 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

An EccentricDigraph Iteration Sequence

An EccentricDigraph Iteration Sequence

An EccentricDigraph Iteration Sequence

G

ED(G)

ED2(G)

ED3(G)

ED4(G)

t=3

p=2

IsomorphismsFor 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 = 2Iso-period = 1

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)

How long can the tail be?

Finite – so there are digraphs that are not eccentric digraphs for any other (di)graph.

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

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?

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

p(G) = 2

Theorem: (Wormald) Almost all digraphs haveiteration sequence period = 2

Period and Iso-period

p(Km Kn) = p(Km,n) = 2, t(Km Kn) = t(Km,n) = 0

Recall

Period and Iso-period

p(Hm Hn) = 2, t(Hm Hn) = 1

Period and Iso-period

1)(,2)( nn CpCp

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

3

33

3

3

3

3

2 3

32

2

2

2

3

3

2

3

3

33

3

3

3

3

2 3

32

2

2

2

3

3

2 K2 K4 K2 K4

K2 C4 K2 C4

R = The Cayley graph with generators(01)(23)(4567) and (56)(78)

A digraph G of order 10 such that p(G) = p'(G) = 4

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

The graph C9 and its iterated eccentric (di)graphs

Eccentric (di)graph period for odd cycles

m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

p(C2m+1) 1 2 3 3 5 6 4 4 9 6 11 10 9 14 5 5

Sequence A003558 in Sloane’s Encyclopedia of Integer Sequences

m 1 2 3 5 6 9 11 14

p(C2m+1) 1 2 3 5 6 9 11 14

Sloane’s A045639, the Queneau Numbers

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

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

Equivalence classes induced by ED for n = 3

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?