Preliminary Lemmas Sample Clauses

Preliminary Lemmas. ‌ We now state several results we will need throughout the argument. We will need to apply a version of the Poisson summation formula.

Preliminary Lemmas. We prove in this section some useful properties and lemmas on reconciliations. The following fact follows directly from the impossibility that all children of a parasite node jump to incomparable host tree nodes. Fact 3. Given any reconciliation q : V (P ) → V (H), for each node u ∈ V (P ):

Preliminary Lemmas. In this section we will prove all preliminary lemmas needed to show pan- cyclicity of 4-connected {K1,3, N(i, j, k)}-free graphs, with i, j, k ≥ 0 and i + j + k = 5. At first, we will consider cycles that do contain chords and afterwards we will consider induced cycles. To show Lemma 4.2 we will follow the basic idea that was used in [13] to show that a 3-connected {T }- free graph, where T is one of N(4, 0, 0), N(3, 1, 0) or P7, is pancyclic. In particular, we use this lemma from [13] to prove the theorem that follows.