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Structure and Properties of Eccentric Digraphs Joint work of Joan Gimbert Universitat de Lleida, Spain Nacho LopezUniversitat de Lleida, Spain Mirka Miller University of Ballarat, Australia Frank Ruskey University of Victoria, Canada Joe Ryan University of Ballarat, Australia

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Eccentric Digraph of a Graph e G (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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A Graph and its Eccentric Digraph

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

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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)ED 2 (G) = G For converse, just change direction of the arrows.

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Symmetric Eccentric Digraphs For G a connected graph ED(G) is symmetric G is self centered (Not true for digraphs See C 4, K 3 K 2 for examples) For G not strongly connected digraph, ED(G) is symmetric G=H 1 H 2 … H k or G=K n →(H 1 H 2 …H k ) Where H 1, H 2 …are strongly connected components

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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

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Eccentric Digraphs and Complements The symmetric case C6C6 ED(C 6 ) = 3K 2 ED 2 (C 6 ) = H 2,3 C 2n ED(C 2n ) = nK 2 ED 2 (C 2n ) = H 2,n The Even Cycle

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Eccentric Digraphs and Complements Construct G – (the reduction of G) by removing all out-arcs of v where out-deg(v) = n-1 GG–G–

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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–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

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An Eccentric Digraph Iteration Sequence

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An Eccentric Digraph Iteration Sequence

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An Eccentric Digraph Iteration Sequence G ED(G) ED 2 (G) ED 3 (G) ED 4 (G) t=3 p=2

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Isomorphisms For every digraph G there exist smallest integer numbers p' > 0 and t' 0 such that ED t ' (G) ED p ' +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

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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)

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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).

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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–G– ED(G – )

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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

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Period and Iso-period p(K m K n ) = p(K m,n ) = 2, t(K m K n ) = t(K m,n ) = 0 Recall

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Period and Iso-period p(H m H n ) = 2, t(H m H n ) = 1

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Period and Iso-period

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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)

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3 3 3 3 3 3 3 3 2 3 3 2 2 2 2 3 3 2

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3 3 3 3 3 3 3 3 2 3 3 2 2 2 2 3 3 2 K 2 K 4 K 2 K 4 K 2 C 4 K 2 C 4

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

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

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The graph C 9 and its iterated eccentric (di)graphs

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Eccentric (di)graph period for odd cycles m12345678910111213141516 p( C 2m+1 ) 1233564496111091455 Sequence A003558 in Sloane’s Encyclopedia of Integer Sequences m1235691114 p( C 2m+1 ) 1235691114 Sloane’s A045639, the Queneau Numbers p( C 2m+1 ) = min{k>1: m(m+1) k-1 = 1 mod(2m+1)} In particular, m = 2 k, p( C 2m+1 ) = k+1

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Equivalence classes induced by ED for n = 3

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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?

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