Classifying Trees with EdgeDeleted

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• Classifying Trees with Edge-Deleted
• Central Appendage Number 2
• Shubhangi Stalder
• University of Wisconsin Waukesha
• Linda Eroh, John Koker,
• Hosien S. Moghadam, Steven J. Winters
• University of Wisconsin Oshkosh

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Basic Definitions

• A graph G is 2-edge-connected if the removal
  of any edge of G never results in a
  disconnected graph.

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• For a vertex v in a 2-edge-connected graph G,
  the edge-deleted eccentricity g(v) of v is
  defined to be the maximum eccentricity of v in
  G e over all edges e of G.
• Example - g(w)9
• and g(x)g(y)10

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• The vertices of G with minimum edge-deleted
  eccentricity are called edge-deleted central
  vertices while the vertices of maximum
  edge-deleted eccentricity are called edge-deleted
  peripheral vertices.
• The subgraph induced by the edge-deleted central
  vertices of G is called the edge-deleted center
  EDC(G) of G and the subgraph induced by the
  edge-deleted peripheral vertices EDP(G) is
  called the edge-deleted periphery.

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• The central appendage number of a graph G is
  the minimum difference V(H) V(G) over all
  graphs H with C(H) ?? G.
• The edge-deleted central appendage number A(G)
  of a graph G is the minimum difference V(H)
  V(G) over all graphs H with EDC(H) ?? G.
  
• A branch of a tree T is a component of
  T V(C(T)).

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Background for Results

• Previously know results for central appendage
  number
• For every graph G, there exists a connected
  graph H such that C(H) ? G.
• 2 ? central appendage number of connected graph ?
  4.
• There are no trees with central appendage number
  3.
• If in a tree all end vertices are equidistant
  from the center, then the central appendage is 2,
  and otherwise 4.

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• Previously know results for A(G) (Koker,
  Moghadam, Stalder, and Winters 2002)
• For every graph G, there exists a 2-
  edge-connected graph H such that EDC(H) ? G.
• 2 ? A(T) ? 3.
• If T is a tree with radius n that satisfies all
  the properties below, then A(T) 2.
• C(T) lt wgt.
• deg(w) ? 4.
• For z ? V(T), if 1 ? dT(z, w) lt n 1, then
  degT z 2, and if dT(w, z) n 1, then degT z
  ? 2.

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Basic Assumptions

• When T is a tree with A(T) 2, let H be a
  graph with V(H) V(T) U x, y and EDC(H)
  T.
• Since g(x) g(y) (call this number k), there
  exists an e ? E(H) such that dH-e(x, y) k.
• Let D be the set of peripheral vertices of T.

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Result 1

• If T is a tree with C(T) ltw, w?gt, then
  A(T) ? 2.
• Proof Let if possible A(T) 2. Now prove some
  general results for all trees with A(T) 2
• g(u) k-1 for all u in T.
• e cannot be an edge of T.
• For peripheral vertices u,v of T, with ux, vy ?
  E(He)
• Shortest u-v path lies entirely in T, and
  dHe(u, v) k 2 diam T, eH-e(v) eH-e(u)
  k-1
• u and v cannot be end vertices of the same
  branch of T.

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• D must contain at least 2 vertices u,v ? V(He)
  with ux, vy ? E(He).
• All end vertices are equidistant from the center
  (k 2 rad(T) 1 when C(T)ltw,w? gt k2 rad(T)
  2 when C(T)ltwgt)
• e xy.
• Since C(T) ltw, w?gt, without loss of
  generality, these results give us that all end
  vertices of w must be adjacent to x and those of
  w? to y. Therefore, dHww?(w, w?) gt k 1 which is
  a contradiction to the fact g(w)k-1, proving
  Result 1 A(T) ? 2.

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Result 2

• Let T be a tree with C(T) ltwgt. Then A(T)
  2 if and only if the following are satisfied
• All end vertices are equidistant from the
  center.
• deg w ? 4.
• For z ? V(T), if 1 ? dT(z, w) lt rad T 1,
  then degT z 2, and if dT(z,w) rad T 1,
  then degTz ? 2.

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• Proof ?(? Koker, Moghadam, Stalder, and Winters,
  2002)
• ?
• The general results necessary for Result 1 give
  us that all end vertices are equidistant from the
  center, and that e xy.
• Show deg w ? 4. (Let if possible deg w lt 4.)
• Show for z ? V(T), if 1 ? dT(z, w) lt n 1,
  then degT z 2, and if dT(w, z) n 1, then
  degT z ? 2.

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• Lemma 1. gH(u) k 1 for all u ? V(T).
• Lemma 2. dHe(x, y) k, then e ? E(T).

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• Lemma 3. If u, v ? V(T) such that ux and vy
  are edges in He, then
• a shortest u-v path in He lies entirely in T
• dHe(u, v) k 1 or k 2
• eHe(u) eHe (v) k 1.