Structure and Properties of Eccentric Digraphs

1. 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

2. 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

3. A Graph and its Eccentric Digraph

4. Eccentric Digraphs and Other Graph and Digraph Operators • Converse • Symmetry (eccentric graph) • Complement

5. 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

6. 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

7. 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

8. 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

9. 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–

10. 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

11. An Eccentric Digraph Iteration Sequence

12. An Eccentric Digraph Iteration Sequence

13. An Eccentric Digraph Iteration Sequence G t=3 ED(G) ED2(G) ED3(G) p=2 ED4(G)

14. 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

15. 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)

16. 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).

17. 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–)

18. 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

19. Period and Iso-period Recall p(Km Kn) = p(Km,n) = 2, t(Km Kn) = t(Km,n) = 0

20. Period and Iso-period p(Hm Hn) = 2, t(Hm Hn) = 1

21. Period and Iso-period

22. 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)

23. 3 3 2 2 3 3 3 2 3 2 3 3 3 2 3 3 2 3

24. 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

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

26. A digraph G of order 10 such that p(G) = p'(G) = 4 and t(G) = t'(G) = 1

27. The graph C9 and its iterated eccentric (di)graphs

28. 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

29. Equivalence classes induced by ED for n = 3

31. 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?