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Broadcast Algorithm Scycle : INPUT: A k-cycle graph Gk where l1 ≥ l2 ≥ ... ≥ lk ≥ 2 and any originator x. OUTPUT: Broadcast time bScycle (x) and scheme of Gk . BROADCAST-SCHEME-Scycle (Gk , l1 ≥ l2 ≥ ... ≥ lk ≥ 2, x) 1. when x = u, u broadcasts along C1 at time unit 1. 2. when x = w, w first informs along the shorter path towards u. 1 u is informed at time d. 2 u informs the cycle with maximum number of uninformed vertices at time d + 1. 3. 1 X0 : It consists of the cycles where there are no informed vertices.

Now, we are going to consider two cases: a) k < lk : This ensures that all the cycles receive the message twice from u. , k + k − 1 + lk −1−(k−1) 2 } = k+ lk +k−2 2 ≤ lk +3k−1 2 b) k ≥ lk : By time k, u has informed at least one vertex in each cycle and each cycle has at most lk − 1 uninformed vertices at time unit k. As a result, it will take at most another lk − 1 time units to inform all the vertices in Gk . Thus, = lk +3k−1 as k > lk . bS (u) ≤ k + lk − 1 = k + 2lk2−2 < k + 2lk2−1 < k + lk +k−1 2 2 lk +3k−1 Thus, for both cases, we get bS (u) ≤ .

3 6 6−δ < 6+5δ 6−δ < Existence of Bounded δ-ηρ-Spine In the next lemma we show that there exists a δ-ηρ-spine Y of T whose |ext(Y )| is bounded by a function of δ. Lemma 3. Given 0 < δ ≤ 1 2 and a spanning tree T of G, there exists a δ-ηρ- spine Y of T satisfying |ext(Y )| ≤ 3 6 2 δ − 11 6 δ +1 . Proof. Consider a minimal δ-ρ-separator Sρ of T and a minimal δ-η-separator Sη of T . If Sρ and Sη have at least one node in common, then define S = Sρ ∪Sη and obviously S is a δ-ηρ-separator. If Sρ and Sη have no nodes in common, then since both are trees, Sη must be included in a component of T − Sρ .

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Algorithms - Sequential, Parallel - A Unified Appr. by R. Miller, L. Boxer

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