+ ['Proof.', 'First let v β V be incident to at least three leaves and suppose there is a minimum power dominating set S of G that does not contain v. If S excludes two or more of the leaves of G incident to v, then those leaves cannot be dominated or forced at any step.', 'Thus, S excludes at most one leaf incident to v, which means S contains at least two leaves β 1 and β 2 incident to v. Then, (S\\{β 1 , β 2 }) βͺ {v} is a smaller power dominating set than S, which is a contradiction.', 'Now consider the case in which v β V is incident to exactly two leaves, β 1 and β 2 , and suppose there is a minimum power dominating set S of G such that {v, β 1 , β 2 } β© S = β
.', 'Then neither β 1 nor β 2 can be dominated or forced at any step, contradicting the assumption that S is a power dominating set.', 'If S is a power dominating set that contains β 1 or β 2 , say β 1 , then (S\\{β 1 }) βͺ {v} is also a power dominating set and has the same cardinality.', 'Applying this to every vertex incident to exactly two leaves produces the minimum power dominating set required by (3).', 'Definition 3.4.', 'Given a graph G = (V, E) and a set X β V , define β r (G, X) as the graph obtained by attaching r leaves to each vertex in X. If X = {v 1 , . . . , v k }, we denote the r leaves attached to vertex v i as β']),
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