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
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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
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 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)
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