Transcript Document

Structure and Properties of
Eccentric Digraphs
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Joan Gimbert
Nacho Lopez
Mirka Miller
Frank Ruskey
Joe Ryan
Joint work of
Universitat de Lleida, Spain
Universitat de Lleida, Spain
University of Ballarat, Australia
University of Victoria, Canada
University of Ballarat, Australia
Eccentric Digraph of a Graph
eG(u) – the eccentricity of a vertex u in a graph G
max
d (u, v)
e(u ) 
v 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
A Graph and its Eccentric Digraph
Eccentric Digraphs and Other Graph
and Digraph Operators
• Converse
• Symmetry (eccentric graph)
• Complement
Eccentric Digraphs and Converse
For converse, just
change direction
of the arrows.
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
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
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 case
The Even Cycle
C6
ED(C6) = 3K2
ED2(C6) = H2,3
C2n
ED(C2n) = nK2
ED2(C2n) = H2,n
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 Eccentric
Digraph
Iteration
Sequence
An Eccentric
Digraph
Iteration
Sequence
An Eccentric
Digraph
Iteration
Sequence
G
ED(G)
t=3
ED2(G)
ED3(G)
p=2
ED4(G)
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
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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
G
G–
ED(G–)
Theorem (Gimbert, Lopez, Miller, R; to appear)
A digraph G is eccentric if and only if
ED(G–) = 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 have
iteration sequence period = 2
Period and Iso-period
Recall
p(Km Kn) = p(Km,n) = 2,
t(Km Kn) = t(Km,n) = 0
Period and Iso-period
p(Hm Hn) = 2,
t(Hm Hn) = 1
Period and Iso-period
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
p(Cn)  2, p(Cn)  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
3
2
3
3
3
2
3
2
3
3
3
2
3
2
3
3
3
2
3
2
3
K2  K4
K2  K4
K2  C4
K2  C4
3
3
2
3
2
3
3
3
2
3
2
3
3
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
p(C2m+1) = min{k>1: m(m+1)k-1 =  1 mod(2m+1)}
In particular, m = 2k, p(C2m+1) = k+1
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
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?